WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            bsort(0(),xs) -> xs
            bsort(S(x'),Cons(x,xs)) -> bsort(x',bubble(x,xs))
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubblesort(xs) -> bsort(len(xs),xs)
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1} / {0/0,Cons/2,False/0,Nil/0,S/1
            ,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,<,bsort,bubble,bubble[Ite][False][Ite],bubblesort
            ,len} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          bsort#(0(),xs) -> c_1()
          bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          bubble#(x,Nil()) -> c_3()
          bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
          len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
          len#(Nil()) -> c_7()
        Weak DPs
          +#(x,S(0())) -> c_8()
          +#(S(0()),y) -> c_9()
          <#(x,0()) -> c_10()
          <#(0(),S(y)) -> c_11()
          <#(S(x),S(y)) -> c_12(<#(x,y))
          bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(0(),xs) -> c_1()
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
            len#(Nil()) -> c_7()
        - Weak DPs:
            +#(x,S(0())) -> c_8()
            +#(S(0()),y) -> c_9()
            <#(x,0()) -> c_10()
            <#(0(),S(y)) -> c_11()
            <#(S(x),S(y)) -> c_12(<#(x,y))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bsort(0(),xs) -> xs
            bsort(S(x'),Cons(x,xs)) -> bsort(x',bubble(x,xs))
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            bubblesort(xs) -> bsort(len(xs),xs)
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(x,S(0())) -> S(x)
          +(S(0()),y) -> S(y)
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          bubble(x,Nil()) -> Cons(x,Nil())
          bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
          bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
          bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
          len(Cons(x,xs)) -> +(S(0()),len(xs))
          len(Nil()) -> 0()
          +#(x,S(0())) -> c_8()
          +#(S(0()),y) -> c_9()
          <#(x,0()) -> c_10()
          <#(0(),S(y)) -> c_11()
          <#(S(x),S(y)) -> c_12(<#(x,y))
          bsort#(0(),xs) -> c_1()
          bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          bubble#(x,Nil()) -> c_3()
          bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
          len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
          len#(Nil()) -> c_7()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(0(),xs) -> c_1()
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
            len#(Nil()) -> c_7()
        - Weak DPs:
            +#(x,S(0())) -> c_8()
            +#(S(0()),y) -> c_9()
            <#(x,0()) -> c_10()
            <#(0(),S(y)) -> c_11()
            <#(S(x),S(y)) -> c_12(<#(x,y))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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
          {1,7}
        by application of
          Pre({1,7}) = {2,5,6}.
        Here rules are labelled as follows:
          1: bsort#(0(),xs) -> c_1()
          2: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          3: bubble#(x,Nil()) -> c_3()
          4: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          5: bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
          6: len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
          7: len#(Nil()) -> c_7()
          8: +#(x,S(0())) -> c_8()
          9: +#(S(0()),y) -> c_9()
          10: <#(x,0()) -> c_10()
          11: <#(0(),S(y)) -> c_11()
          12: <#(S(x),S(y)) -> c_12(<#(x,y))
          13: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          14: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
        - Weak DPs:
            +#(x,S(0())) -> c_8()
            +#(S(0()),y) -> c_9()
            <#(x,0()) -> c_10()
            <#(0(),S(y)) -> c_11()
            <#(S(x),S(y)) -> c_12(<#(x,y))
            bsort#(0(),xs) -> c_1()
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            len#(Nil()) -> c_7()
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_1 bsort#(0(),xs) -> c_1():11
             -->_2 bubble#(x,Nil()) -> c_3():2
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:S:bubble#(x,Nil()) -> c_3()
             
          
          3:S:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):13
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):12
             -->_2 <#(S(x),S(y)) -> c_12(<#(x,y)):10
             -->_2 <#(0(),S(y)) -> c_11():9
             -->_2 <#(x,0()) -> c_10():8
          
          4:S:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_2 len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs)):5
             -->_2 len#(Nil()) -> c_7():14
             -->_1 bsort#(0(),xs) -> c_1():11
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          5:S:len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
             -->_2 len#(Nil()) -> c_7():14
             -->_1 +#(S(0()),y) -> c_9():7
             -->_1 +#(x,S(0())) -> c_8():6
             -->_2 len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs)):5
          
          6:W:+#(x,S(0())) -> c_8()
             
          
          7:W:+#(S(0()),y) -> c_9()
             
          
          8:W:<#(x,0()) -> c_10()
             
          
          9:W:<#(0(),S(y)) -> c_11()
             
          
          10:W:<#(S(x),S(y)) -> c_12(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_12(<#(x,y)):10
             -->_1 <#(0(),S(y)) -> c_11():9
             -->_1 <#(x,0()) -> c_10():8
          
          11:W:bsort#(0(),xs) -> c_1()
             
          
          12:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_1 bubble#(x,Nil()) -> c_3():2
          
          13:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_1 bubble#(x,Nil()) -> c_3():2
          
          14:W:len#(Nil()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: +#(x,S(0())) -> c_8()
          7: +#(S(0()),y) -> c_9()
          14: len#(Nil()) -> c_7()
          11: bsort#(0(),xs) -> c_1()
          10: <#(S(x),S(y)) -> c_12(<#(x,y))
          8: <#(x,0()) -> c_10()
          9: <#(0(),S(y)) -> c_11()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/2,c_5/2,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_2 bubble#(x,Nil()) -> c_3():2
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:S:bubble#(x,Nil()) -> c_3()
             
          
          3:S:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):13
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):12
          
          4:S:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_2 len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs)):5
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          5:S:len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs))
             -->_2 len#(Cons(x,xs)) -> c_6(+#(S(0()),len(xs)),len#(xs)):5
          
          12:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_1 bubble#(x,Nil()) -> c_3():2
          
          13:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):3
             -->_1 bubble#(x,Nil()) -> c_3():2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          len#(Cons(x,xs)) -> c_6(len#(xs))
* Step 6: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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:
              bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
              bubble#(x,Nil()) -> c_3()
              bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          - Weak DPs:
              bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
              bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
              bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
              len#(Cons(x,xs)) -> c_6(len#(xs))
          - Weak TRS:
              +(x,S(0())) -> S(x)
              +(S(0()),y) -> S(y)
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              bubble(x,Nil()) -> Cons(x,Nil())
              bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
              bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
              len(Cons(x,xs)) -> +(S(0()),len(xs))
              len(Nil()) -> 0()
          - Signature:
              {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
              ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
              ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#
              ,bubblesort#,len#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
              len#(Cons(x,xs)) -> c_6(len#(xs))
          - Weak DPs:
              bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
              bubble#(x,Nil()) -> c_3()
              bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
              bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
              bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          - Weak TRS:
              +(x,S(0())) -> S(x)
              +(S(0()),y) -> S(y)
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              bubble(x,Nil()) -> Cons(x,Nil())
              bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
              bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
              len(Cons(x,xs)) -> +(S(0()),len(xs))
              len(Nil()) -> 0()
          - Signature:
              {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
              ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
              ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#
              ,bubblesort#,len#} and constructors {0,Cons,False,Nil,S,True}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
             -->_2 bubble#(x,Nil()) -> c_3():2
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:S:bubble#(x,Nil()) -> c_3()
             
          
          3:S:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):7
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):6
          
          4:W:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
             -->_2 len#(Cons(x,xs)) -> c_6(len#(xs)):5
          
          5:W:len#(Cons(x,xs)) -> c_6(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):5
          
          6:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x,Nil()) -> c_3():2
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
          
          7:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x,Nil()) -> c_3():2
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: len#(Cons(x,xs)) -> c_6(len#(xs))
** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
             -->_2 bubble#(x,Nil()) -> c_3():2
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:S:bubble#(x,Nil()) -> c_3()
             
          
          3:S:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):7
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):6
          
          4:W:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          6:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x,Nil()) -> c_3():2
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
          
          7:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x,Nil()) -> c_3():2
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
** Step 6.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          
        Consider the set of all dependency pairs
          1: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          2: bubble#(x,Nil()) -> c_3()
          3: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          4: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          5: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          6: bubblesort#(xs) -> c_5(bsort#(len(xs),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
          {1,6}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
*** Step 6.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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_2) = {1,2},
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_13) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {+,len,+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#,len#}
        TcT has computed the following interpretation:
                                 p(+) = [1] x1 + [1] x2 + [0]
                                 p(0) = [0]                  
                                 p(<) = [0]                  
                              p(Cons) = [1] x2 + [1]         
                             p(False) = [0]                  
                               p(Nil) = [2]                  
                                 p(S) = [1] x1 + [1]         
                              p(True) = [1]                  
                             p(bsort) = [0]                  
                            p(bubble) = [0]                  
           p(bubble[Ite][False][Ite]) = [4]                  
                        p(bubblesort) = [0]                  
                               p(len) = [1] x1 + [0]         
                                p(+#) = [1] x2 + [0]         
                                p(<#) = [0]                  
                            p(bsort#) = [1] x1 + [0]         
                           p(bubble#) = [0]                  
          p(bubble[Ite][False][Ite]#) = [0]                  
                       p(bubblesort#) = [2] x1 + [4]         
                              p(len#) = [4] x1 + [4]         
                               p(c_1) = [1]                  
                               p(c_2) = [1] x1 + [4] x2 + [0]
                               p(c_3) = [0]                  
                               p(c_4) = [2] x1 + [0]         
                               p(c_5) = [2] x1 + [4]         
                               p(c_6) = [1] x1 + [1]         
                               p(c_7) = [0]                  
                               p(c_8) = [4]                  
                               p(c_9) = [1]                  
                              p(c_10) = [1]                  
                              p(c_11) = [2]                  
                              p(c_12) = [0]                  
                              p(c_13) = [4] x1 + [0]         
                              p(c_14) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        bsort#(S(x'),Cons(x,xs)) = [1] x' + [1]                              
                                 > [1] x' + [0]                              
                                 = c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
        
        
        Following rules are (at-least) weakly oriented:
                                       bubble#(x,Nil()) =  [0]                                                 
                                                        >= [0]                                                 
                                                        =  c_3()                                               
        
                                 bubble#(x',Cons(x,xs)) =  [0]                                                 
                                                        >= [0]                                                 
                                                        =  c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [0]                                                 
                                                        >= [0]                                                 
                                                        =  c_13(bubble#(x',xs))                                
        
         bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) =  [0]                                                 
                                                        >= [0]                                                 
                                                        =  c_14(bubble#(x,xs))                                 
        
                                        bubblesort#(xs) =  [2] xs + [4]                                        
                                                        >= [2] xs + [4]                                        
                                                        =  c_5(bsort#(len(xs),xs))                             
        
                                            +(x,S(0())) =  [1] x + [1]                                         
                                                        >= [1] x + [1]                                         
                                                        =  S(x)                                                
        
                                            +(S(0()),y) =  [1] y + [1]                                         
                                                        >= [1] y + [1]                                         
                                                        =  S(y)                                                
        
                                        len(Cons(x,xs)) =  [1] xs + [1]                                        
                                                        >= [1] xs + [1]                                        
                                                        =  +(S(0()),len(xs))                                   
        
                                             len(Nil()) =  [2]                                                 
                                                        >= [0]                                                 
                                                        =  0()                                                 
        
*** Step 6.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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: bubble#(x,Nil()) -> c_3()
          
        Consider the set of all dependency pairs
          1: bubble#(x,Nil()) -> c_3()
          2: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          3: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          4: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          5: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          6: bubblesort#(xs) -> c_5(bsort#(len(xs),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
          {1,6}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 6.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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_2) = {1,2},
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_13) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {+,len,+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#,len#}
        TcT has computed the following interpretation:
                                 p(+) = [1] x1 + [1] x2 + [0]         
                                 p(0) = [0]                           
                                 p(<) = [0]                           
                              p(Cons) = [1] x2 + [1]                  
                             p(False) = [0]                           
                               p(Nil) = [0]                           
                                 p(S) = [1] x1 + [2]                  
                              p(True) = [0]                           
                             p(bsort) = [1]                           
                            p(bubble) = [0]                           
           p(bubble[Ite][False][Ite]) = [2] x1 + [7] x2 + [1] x3 + [3]
                        p(bubblesort) = [1]                           
                               p(len) = [4] x1 + [0]                  
                                p(+#) = [4] x1 + [2] x2 + [0]         
                                p(<#) = [1]                           
                            p(bsort#) = [1] x1 + [4]                  
                           p(bubble#) = [1]                           
          p(bubble[Ite][False][Ite]#) = [1]                           
                       p(bubblesort#) = [4] x1 + [7]                  
                              p(len#) = [2] x1 + [0]                  
                               p(c_1) = [0]                           
                               p(c_2) = [1] x1 + [1] x2 + [1]         
                               p(c_3) = [0]                           
                               p(c_4) = [1] x1 + [0]                  
                               p(c_5) = [1] x1 + [3]                  
                               p(c_6) = [0]                           
                               p(c_7) = [1]                           
                               p(c_8) = [1]                           
                               p(c_9) = [0]                           
                              p(c_10) = [1]                           
                              p(c_11) = [4]                           
                              p(c_12) = [0]                           
                              p(c_13) = [1] x1 + [0]                  
                              p(c_14) = [1] x1 + [0]                  
        
        Following rules are strictly oriented:
        bubble#(x,Nil()) = [1]  
                         > [0]  
                         = c_3()
        
        
        Following rules are (at-least) weakly oriented:
                               bsort#(S(x'),Cons(x,xs)) =  [1] x' + [6]                                        
                                                        >= [1] x' + [6]                                        
                                                        =  c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))          
        
                                 bubble#(x',Cons(x,xs)) =  [1]                                                 
                                                        >= [1]                                                 
                                                        =  c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [1]                                                 
                                                        >= [1]                                                 
                                                        =  c_13(bubble#(x',xs))                                
        
         bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) =  [1]                                                 
                                                        >= [1]                                                 
                                                        =  c_14(bubble#(x,xs))                                 
        
                                        bubblesort#(xs) =  [4] xs + [7]                                        
                                                        >= [4] xs + [7]                                        
                                                        =  c_5(bsort#(len(xs),xs))                             
        
                                            +(x,S(0())) =  [1] x + [2]                                         
                                                        >= [1] x + [2]                                         
                                                        =  S(x)                                                
        
                                            +(S(0()),y) =  [1] y + [2]                                         
                                                        >= [1] y + [2]                                         
                                                        =  S(y)                                                
        
                                        len(Cons(x,xs)) =  [4] xs + [4]                                        
                                                        >= [4] xs + [2]                                        
                                                        =  +(S(0()),len(xs))                                   
        
                                             len(Nil()) =  [0]                                                 
                                                        >= [0]                                                 
                                                        =  0()                                                 
        
**** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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 6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):5
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):4
          
          2:W:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x,Nil()) -> c_3():3
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):2
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
          3:W:bubble#(x,Nil()) -> c_3()
             
          
          4:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x,Nil()) -> c_3():3
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
          5:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x,Nil()) -> c_3():3
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
          6:W:bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: bubble#(x,Nil()) -> c_3()
**** Step 6.a:3.b:1.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          
        Consider the set of all dependency pairs
          1: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          2: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          4: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          5: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          6: bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,4,5,6}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
***** Step 6.a:3.b:1.b:2.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_2) = {1,2},
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_13) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {+,bubble,bubble[Ite][False][Ite],len,+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#,len#}
        TcT has computed the following interpretation:
                                 p(+) = 2*x1 + x2                  
                                 p(0) = 0                          
                                 p(<) = 0                          
                              p(Cons) = 2 + x2                     
                             p(False) = 0                          
                               p(Nil) = 2                          
                                 p(S) = 1 + x1                     
                              p(True) = 0                          
                             p(bsort) = x2 + 2*x2^2                
                            p(bubble) = 2 + x2                     
           p(bubble[Ite][False][Ite]) = 2 + x3                     
                        p(bubblesort) = 2*x1                       
                               p(len) = x1                         
                                p(+#) = x1 + x1*x2                 
                                p(<#) = 1 + x1 + x1^2 + 2*x2 + x2^2
                            p(bsort#) = 2*x1*x2 + x1^2             
                           p(bubble#) = 2 + 2*x2                   
          p(bubble[Ite][False][Ite]#) = 2*x3                       
                       p(bubblesort#) = 3 + 3*x1^2                 
                              p(len#) = x1                         
                               p(c_1) = 1                          
                               p(c_2) = 3 + x1 + x2                
                               p(c_3) = 2                          
                               p(c_4) = x1                         
                               p(c_5) = x1                         
                               p(c_6) = x1                         
                               p(c_7) = 0                          
                               p(c_8) = 2                          
                               p(c_9) = 2                          
                              p(c_10) = 1                          
                              p(c_11) = 0                          
                              p(c_12) = x1                         
                              p(c_13) = 2 + x1                     
                              p(c_14) = 1 + x1                     
        
        Following rules are strictly oriented:
        bubble#(x',Cons(x,xs)) = 6 + 2*xs                                            
                               > 4 + 2*xs                                            
                               = c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
                               bsort#(S(x'),Cons(x,xs)) =  5 + 6*x' + 2*x'*xs + x'^2 + 2*xs              
                                                        >= 5 + 4*x' + 2*x'*xs + x'^2 + 2*xs              
                                                        =  c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))    
        
        bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  4 + 2*xs                                      
                                                        >= 4 + 2*xs                                      
                                                        =  c_13(bubble#(x',xs))                          
        
         bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) =  4 + 2*xs                                      
                                                        >= 3 + 2*xs                                      
                                                        =  c_14(bubble#(x,xs))                           
        
                                        bubblesort#(xs) =  3 + 3*xs^2                                    
                                                        >= 3*xs^2                                        
                                                        =  c_5(bsort#(len(xs),xs))                       
        
                                            +(x,S(0())) =  1 + 2*x                                       
                                                        >= 1 + x                                         
                                                        =  S(x)                                          
        
                                            +(S(0()),y) =  2 + y                                         
                                                        >= 1 + y                                         
                                                        =  S(y)                                          
        
                                        bubble(x,Nil()) =  4                                             
                                                        >= 4                                             
                                                        =  Cons(x,Nil())                                 
        
                                  bubble(x',Cons(x,xs)) =  4 + xs                                        
                                                        >= 4 + xs                                        
                                                        =  bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
        
         bubble[Ite][False][Ite](False(),x',Cons(x,xs)) =  4 + xs                                        
                                                        >= 4 + xs                                        
                                                        =  Cons(x,bubble(x',xs))                         
        
          bubble[Ite][False][Ite](True(),x',Cons(x,xs)) =  4 + xs                                        
                                                        >= 4 + xs                                        
                                                        =  Cons(x',bubble(x,xs))                         
        
                                        len(Cons(x,xs)) =  2 + xs                                        
                                                        >= 2 + xs                                        
                                                        =  +(S(0()),len(xs))                             
        
                                             len(Nil()) =  2                                             
                                                        >= 0                                             
                                                        =  0()                                           
        
***** Step 6.a:3.b:1.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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 6.a:3.b:1.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:W:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):4
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):3
          
          3:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
          
          4:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
          
          5:W:bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: bubblesort#(xs) -> c_5(bsort#(len(xs),xs))
          1: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          2: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          4: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          3: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
***** Step 6.a:3.b:1.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
            bubble#(x,Nil()) -> c_3()
            bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
            bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):3
             -->_2 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
          2:S:len#(Cons(x,xs)) -> c_6(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
          3:W:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_2 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):5
             -->_2 bubble#(x,Nil()) -> c_3():4
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):3
          
          4:W:bubble#(x,Nil()) -> c_3()
             
          
          5:W:bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs)):7
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):6
          
          6:W:bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):5
             -->_1 bubble#(x,Nil()) -> c_3():4
          
          7:W:bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):5
             -->_1 bubble#(x,Nil()) -> c_3():4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          5: bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          7: bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) -> c_14(bubble#(x,xs))
          6: bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs))
          4: bubble#(x,Nil()) -> c_3()
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_2 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
          2:S:len#(Cons(x,xs)) -> c_6(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bubblesort#(xs) -> c_5(len#(xs))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Weak TRS:
            +(x,S(0())) -> S(x)
            +(S(0()),y) -> S(y)
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            bubble(x,Nil()) -> Cons(x,Nil())
            bubble(x',Cons(x,xs)) -> bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            bubble[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,bubble(x',xs))
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) -> Cons(x',bubble(x,xs))
            len(Cons(x,xs)) -> +(S(0()),len(xs))
            len(Nil()) -> 0()
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          bubblesort#(xs) -> c_5(len#(xs))
          len#(Cons(x,xs)) -> c_6(len#(xs))
** Step 6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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: bubblesort#(xs) -> c_5(len#(xs))
          2: len#(Cons(x,xs)) -> c_6(len#(xs))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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_5) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#,len#}
        TcT has computed the following interpretation:
                                 p(+) = [0]                  
                                 p(0) = [0]                  
                                 p(<) = [0]                  
                              p(Cons) = [1] x1 + [1] x2 + [1]
                             p(False) = [0]                  
                               p(Nil) = [0]                  
                                 p(S) = [1] x1 + [0]         
                              p(True) = [0]                  
                             p(bsort) = [1] x1 + [2] x2 + [1]
                            p(bubble) = [1]                  
           p(bubble[Ite][False][Ite]) = [2] x2 + [1]         
                        p(bubblesort) = [1]                  
                               p(len) = [1] x1 + [1]         
                                p(+#) = [8] x1 + [0]         
                                p(<#) = [2]                  
                            p(bsort#) = [1] x2 + [1]         
                           p(bubble#) = [1] x1 + [2]         
          p(bubble[Ite][False][Ite]#) = [1] x1 + [2] x3 + [1]
                       p(bubblesort#) = [8] x1 + [12]        
                              p(len#) = [2] x1 + [2]         
                               p(c_1) = [1]                  
                               p(c_2) = [1] x1 + [1]         
                               p(c_3) = [8]                  
                               p(c_4) = [1] x1 + [1]         
                               p(c_5) = [2] x1 + [1]         
                               p(c_6) = [1] x1 + [1]         
                               p(c_7) = [2]                  
                               p(c_8) = [0]                  
                               p(c_9) = [1]                  
                              p(c_10) = [1]                  
                              p(c_11) = [0]                  
                              p(c_12) = [2]                  
                              p(c_13) = [2] x1 + [1]         
                              p(c_14) = [1]                  
        
        Following rules are strictly oriented:
         bubblesort#(xs) = [8] xs + [12]       
                         > [4] xs + [5]        
                         = c_5(len#(xs))       
        
        len#(Cons(x,xs)) = [2] x + [2] xs + [4]
                         > [2] xs + [3]        
                         = c_6(len#(xs))       
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bubblesort#(xs) -> c_5(len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bubblesort#(xs) -> c_5(len#(xs))
            len#(Cons(x,xs)) -> c_6(len#(xs))
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:bubblesort#(xs) -> c_5(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
          2:W:len#(Cons(x,xs)) -> c_6(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: bubblesort#(xs) -> c_5(len#(xs))
          2: len#(Cons(x,xs)) -> c_6(len#(xs))
*** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {+/2,</2,bsort/2,bubble/2,bubble[Ite][False][Ite]/3,bubblesort/1,len/1,+#/2,<#/2,bsort#/2,bubble#/2
            ,bubble[Ite][False][Ite]#/3,bubblesort#/1,len#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0
            ,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#
            ,len#} 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(?,O(n^2))