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: 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)
            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:
        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 3: 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)
            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:
        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 4: 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)
            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:
        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 5: UsableRules 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)
            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_4/1,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:
        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()
          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))
          bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
          len#(Cons(x,xs)) -> c_6(len#(xs))
* Step 6: DecomposeDG 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:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
          bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
          len#(Cons(x,xs)) -> c_6(len#(xs))
        and a lower component
          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))
        Further, following extension rules are added to the lower component.
          bsort#(S(x'),Cons(x,xs)) -> bsort#(x',bubble(x,xs))
          bsort#(S(x'),Cons(x,xs)) -> bubble#(x,xs)
          bubblesort#(xs) -> bsort#(len(xs),xs)
          bubblesort#(xs) -> len#(xs)
          len#(Cons(x,xs)) -> len#(xs)
** Step 6.a:1: SimplifyRHS 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))
            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:bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          2:S:bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
             -->_2 len#(Cons(x,xs)) -> c_6(len#(xs)):3
             -->_1 bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)),bubble#(x,xs)):1
          
          3:S:len#(Cons(x,xs)) -> c_6(len#(xs))
             -->_1 len#(Cons(x,xs)) -> c_6(len#(xs)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,xs)))
** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',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_2/1,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {2},
            uargs(Cons) = {2},
            uargs(bubble[Ite][False][Ite]) = {1},
            uargs(bsort#) = {1,2},
            uargs(c_2) = {1},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          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 + [3]         
                                p(True) = [0]                  
                               p(bsort) = [1] x1 + [1]         
                              p(bubble) = [1] x2 + [1]         
             p(bubble[Ite][False][Ite]) = [1] x1 + [1] x3 + [1]
                          p(bubblesort) = [4] x1 + [2]         
                                 p(len) = [3] x1 + [0]         
                                  p(+#) = [0]                  
                                  p(<#) = [0]                  
                              p(bsort#) = [1] x1 + [1] x2 + [0]
                             p(bubble#) = [0]                  
            p(bubble[Ite][False][Ite]#) = [0]                  
                         p(bubblesort#) = [5] x1 + [0]         
                                p(len#) = [1] x1 + [0]         
                                 p(c_1) = [0]                  
                                 p(c_2) = [1] x1 + [1]         
                                 p(c_3) = [1]                  
                                 p(c_4) = [1]                  
                                 p(c_5) = [1] x1 + [1] x2 + [0]
                                 p(c_6) = [1] x1 + [0]         
                                 p(c_7) = [0]                  
                                 p(c_8) = [0]                  
                                 p(c_9) = [0]                  
                                p(c_10) = [0]                  
                                p(c_11) = [0]                  
                                p(c_12) = [0]                  
                                p(c_13) = [0]                  
                                p(c_14) = [0]                  
          
          Following rules are strictly oriented:
          bsort#(S(x'),Cons(x,xs)) = [1] x' + [1] xs + [4]       
                                   > [1] x' + [1] xs + [2]       
                                   = c_2(bsort#(x',bubble(x,xs)))
          
                  len#(Cons(x,xs)) = [1] xs + [1]                
                                   > [1] xs + [0]                
                                   = c_6(len#(xs))               
          
          
          Following rules are (at-least) weakly oriented:
                                         bubblesort#(xs) =  [5] xs + [0]                                  
                                                         >= [5] xs + [0]                                  
                                                         =  c_5(bsort#(len(xs),xs),len#(xs))              
          
                                             +(x,S(0())) =  [1] x + [3]                                   
                                                         >= [1] x + [3]                                   
                                                         =  S(x)                                          
          
                                             +(S(0()),y) =  [1] y + [3]                                   
                                                         >= [1] y + [3]                                   
                                                         =  S(y)                                          
          
                                                <(x,0()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  False()                                       
          
                                             <(0(),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  True()                                        
          
                                            <(S(x),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  <(x,y)                                        
          
                                         bubble(x,Nil()) =  [1]                                           
                                                         >= [1]                                           
                                                         =  Cons(x,Nil())                                 
          
                                   bubble(x',Cons(x,xs)) =  [1] xs + [2]                                  
                                                         >= [1] xs + [2]                                  
                                                         =  bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
          
          bubble[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] xs + [2]                                  
                                                         >= [1] xs + [2]                                  
                                                         =  Cons(x,bubble(x',xs))                         
          
           bubble[Ite][False][Ite](True(),x',Cons(x,xs)) =  [1] xs + [2]                                  
                                                         >= [1] xs + [2]                                  
                                                         =  Cons(x',bubble(x,xs))                         
          
                                         len(Cons(x,xs)) =  [3] xs + [3]                                  
                                                         >= [3] xs + [3]                                  
                                                         =  +(S(0()),len(xs))                             
          
                                              len(Nil()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  0()                                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            bubblesort#(xs) -> c_5(bsort#(len(xs),xs),len#(xs))
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',bubble(x,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_2/1,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {2},
            uargs(Cons) = {2},
            uargs(bubble[Ite][False][Ite]) = {1},
            uargs(bsort#) = {1,2},
            uargs(c_2) = {1},
            uargs(c_5) = {1,2},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                                   p(+) = [1] x1 + [1] x2 + [0]
                                   p(0) = [0]                  
                                   p(<) = [0]                  
                                p(Cons) = [1] x2 + [0]         
                               p(False) = [0]                  
                                 p(Nil) = [0]                  
                                   p(S) = [1] x1 + [0]         
                                p(True) = [0]                  
                               p(bsort) = [0]                  
                              p(bubble) = [1] x2 + [0]         
             p(bubble[Ite][False][Ite]) = [1] x1 + [1] x3 + [0]
                          p(bubblesort) = [0]                  
                                 p(len) = [0]                  
                                  p(+#) = [0]                  
                                  p(<#) = [0]                  
                              p(bsort#) = [1] x1 + [1] x2 + [0]
                             p(bubble#) = [0]                  
            p(bubble[Ite][False][Ite]#) = [1] x1 + [4] x3 + [1]
                         p(bubblesort#) = [1] x1 + [5]         
                                p(len#) = [4]                  
                                 p(c_1) = [0]                  
                                 p(c_2) = [1] x1 + [0]         
                                 p(c_3) = [0]                  
                                 p(c_4) = [0]                  
                                 p(c_5) = [1] x1 + [1] x2 + [0]
                                 p(c_6) = [1] x1 + [0]         
                                 p(c_7) = [0]                  
                                 p(c_8) = [0]                  
                                 p(c_9) = [0]                  
                                p(c_10) = [0]                  
                                p(c_11) = [0]                  
                                p(c_12) = [0]                  
                                p(c_13) = [0]                  
                                p(c_14) = [0]                  
          
          Following rules are strictly oriented:
          bubblesort#(xs) = [1] xs + [5]                    
                          > [1] xs + [4]                    
                          = c_5(bsort#(len(xs),xs),len#(xs))
          
          
          Following rules are (at-least) weakly oriented:
                                bsort#(S(x'),Cons(x,xs)) =  [1] x' + [1] xs + [0]                         
                                                         >= [1] x' + [1] xs + [0]                         
                                                         =  c_2(bsort#(x',bubble(x,xs)))                  
          
                                        len#(Cons(x,xs)) =  [4]                                           
                                                         >= [4]                                           
                                                         =  c_6(len#(xs))                                 
          
                                             +(x,S(0())) =  [1] x + [0]                                   
                                                         >= [1] x + [0]                                   
                                                         =  S(x)                                          
          
                                             +(S(0()),y) =  [1] y + [0]                                   
                                                         >= [1] y + [0]                                   
                                                         =  S(y)                                          
          
                                                <(x,0()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  False()                                       
          
                                             <(0(),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  True()                                        
          
                                            <(S(x),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  <(x,y)                                        
          
                                         bubble(x,Nil()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  Cons(x,Nil())                                 
          
                                   bubble(x',Cons(x,xs)) =  [1] xs + [0]                                  
                                                         >= [1] xs + [0]                                  
                                                         =  bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
          
          bubble[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] xs + [0]                                  
                                                         >= [1] xs + [0]                                  
                                                         =  Cons(x,bubble(x',xs))                         
          
           bubble[Ite][False][Ite](True(),x',Cons(x,xs)) =  [1] xs + [0]                                  
                                                         >= [1] xs + [0]                                  
                                                         =  Cons(x',bubble(x,xs))                         
          
                                         len(Cons(x,xs)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  +(S(0()),len(xs))                             
          
                                              len(Nil()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  0()                                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> c_2(bsort#(x',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_2/1,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:
        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:
            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)) -> bsort#(x',bubble(x,xs))
            bsort#(S(x'),Cons(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) -> bsort#(len(xs),xs)
            bubblesort#(xs) -> len#(xs)
            len#(Cons(x,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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:bubble#(x,Nil()) -> c_3()
             
          
          2: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)):6
             -->_1 bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_13(bubble#(x',xs)):5
          
          3:W:bsort#(S(x'),Cons(x,xs)) -> bsort#(x',bubble(x,xs))
             -->_1 bsort#(S(x'),Cons(x,xs)) -> bubble#(x,xs):4
             -->_1 bsort#(S(x'),Cons(x,xs)) -> bsort#(x',bubble(x,xs)):3
          
          4:W:bsort#(S(x'),Cons(x,xs)) -> bubble#(x,xs)
             -->_1 bubble#(x',Cons(x,xs)) -> c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
             -->_1 bubble#(x,Nil()) -> c_3():1
          
          5: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
             -->_1 bubble#(x,Nil()) -> c_3():1
          
          6: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
             -->_1 bubble#(x,Nil()) -> c_3():1
          
          7:W:bubblesort#(xs) -> bsort#(len(xs),xs)
             -->_1 bsort#(S(x'),Cons(x,xs)) -> bubble#(x,xs):4
             -->_1 bsort#(S(x'),Cons(x,xs)) -> bsort#(x',bubble(x,xs)):3
          
          8:W:bubblesort#(xs) -> len#(xs)
             -->_1 len#(Cons(x,xs)) -> len#(xs):9
          
          9:W:len#(Cons(x,xs)) -> len#(xs)
             -->_1 len#(Cons(x,xs)) -> len#(xs):9
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: bubblesort#(xs) -> len#(xs)
          9: len#(Cons(x,xs)) -> len#(xs)
** Step 6.b:2: WeightGap 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)) -> bsort#(x',bubble(x,xs))
            bsort#(S(x'),Cons(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) -> 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_5/2,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:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {2},
            uargs(Cons) = {2},
            uargs(bubble[Ite][False][Ite]) = {1},
            uargs(bsort#) = {1,2},
            uargs(bubble[Ite][False][Ite]#) = {1},
            uargs(c_4) = {1},
            uargs(c_13) = {1},
            uargs(c_14) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                                   p(+) = [4] x1 + [1] x2 + [0]
                                   p(0) = [0]                  
                                   p(<) = [0]                  
                                p(Cons) = [1] x2 + [0]         
                               p(False) = [0]                  
                                 p(Nil) = [0]                  
                                   p(S) = [1] x1 + [0]         
                                p(True) = [0]                  
                               p(bsort) = [1] x2 + [4]         
                              p(bubble) = [0]                  
             p(bubble[Ite][False][Ite]) = [1] x1 + [0]         
                          p(bubblesort) = [1]                  
                                 p(len) = [0]                  
                                  p(+#) = [1] x1 + [4] x2 + [2]
                                  p(<#) = [4] x1 + [2]         
                              p(bsort#) = [1] x1 + [1] x2 + [4]
                             p(bubble#) = [2]                  
            p(bubble[Ite][False][Ite]#) = [1] x1 + [2]         
                         p(bubblesort#) = [1] x1 + [5]         
                                p(len#) = [1] x1 + [0]         
                                 p(c_1) = [1]                  
                                 p(c_2) = [1] x1 + [4] x2 + [4]
                                 p(c_3) = [0]                  
                                 p(c_4) = [1] x1 + [5]         
                                 p(c_5) = [0]                  
                                 p(c_6) = [1] x1 + [2]         
                                 p(c_7) = [4]                  
                                 p(c_8) = [0]                  
                                 p(c_9) = [4]                  
                                p(c_10) = [2]                  
                                p(c_11) = [0]                  
                                p(c_12) = [0]                  
                                p(c_13) = [1] x1 + [0]         
                                p(c_14) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          bubble#(x,Nil()) = [2]  
                           > [0]  
                           = c_3()
          
          
          Following rules are (at-least) weakly oriented:
                                 bsort#(S(x'),Cons(x,xs)) =  [1] x' + [1] xs + [4]                               
                                                          >= [1] x' + [4]                                        
                                                          =  bsort#(x',bubble(x,xs))                             
          
                                 bsort#(S(x'),Cons(x,xs)) =  [1] x' + [1] xs + [4]                               
                                                          >= [2]                                                 
                                                          =  bubble#(x,xs)                                       
          
                                   bubble#(x',Cons(x,xs)) =  [2]                                                 
                                                          >= [7]                                                 
                                                          =  c_4(bubble[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          
          bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [2]                                                 
                                                          >= [2]                                                 
                                                          =  c_13(bubble#(x',xs))                                
          
           bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) =  [2]                                                 
                                                          >= [2]                                                 
                                                          =  c_14(bubble#(x,xs))                                 
          
                                          bubblesort#(xs) =  [1] xs + [5]                                        
                                                          >= [1] xs + [4]                                        
                                                          =  bsort#(len(xs),xs)                                  
          
                                              +(x,S(0())) =  [4] x + [0]                                         
                                                          >= [1] x + [0]                                         
                                                          =  S(x)                                                
          
                                              +(S(0()),y) =  [1] y + [0]                                         
                                                          >= [1] y + [0]                                         
                                                          =  S(y)                                                
          
                                                 <(x,0()) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  False()                                             
          
                                              <(0(),S(y)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  True()                                              
          
                                             <(S(x),S(y)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  <(x,y)                                              
          
                                          bubble(x,Nil()) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  Cons(x,Nil())                                       
          
                                    bubble(x',Cons(x,xs)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))      
          
           bubble[Ite][False][Ite](False(),x',Cons(x,xs)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  Cons(x,bubble(x',xs))                               
          
            bubble[Ite][False][Ite](True(),x',Cons(x,xs)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  Cons(x',bubble(x,xs))                               
          
                                          len(Cons(x,xs)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  +(S(0()),len(xs))                                   
          
                                               len(Nil()) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  0()                                                 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.b:3: NaturalMI WORST_CASE(?,O(n^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)) -> bsort#(x',bubble(x,xs))
            bsort#(S(x'),Cons(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) -> 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_5/2,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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_13) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {<,bubble,bubble[Ite][False][Ite],+#,<#,bsort#,bubble#,bubble[Ite][False][Ite]#,bubblesort#,len#}
        TcT has computed the following interpretation:
                                 p(+) = [6] x1 + [1]         
                                 p(0) = [0]                  
                                 p(<) = [1]                  
                              p(Cons) = [1] x2 + [4]         
                             p(False) = [1]                  
                               p(Nil) = [1]                  
                                 p(S) = [2]                  
                              p(True) = [1]                  
                             p(bsort) = [4] x1 + [2] x2 + [1]
                            p(bubble) = [1] x2 + [4]         
           p(bubble[Ite][False][Ite]) = [4] x1 + [1] x3 + [0]
                        p(bubblesort) = [0]                  
                               p(len) = [2] x1 + [1]         
                                p(+#) = [1] x1 + [2] x2 + [0]
                                p(<#) = [4] x2 + [2]         
                            p(bsort#) = [2] x2 + [0]         
                           p(bubble#) = [2] x2 + [6]         
          p(bubble[Ite][False][Ite]#) = [4] x1 + [2] x3 + [0]
                       p(bubblesort#) = [4] x1 + [4]         
                              p(len#) = [0]                  
                               p(c_1) = [0]                  
                               p(c_2) = [4] x1 + [1] x2 + [4]
                               p(c_3) = [4]                  
                               p(c_4) = [1] x1 + [0]         
                               p(c_5) = [2]                  
                               p(c_6) = [1]                  
                               p(c_7) = [1]                  
                               p(c_8) = [0]                  
                               p(c_9) = [0]                  
                              p(c_10) = [0]                  
                              p(c_11) = [1]                  
                              p(c_12) = [1] x1 + [1]         
                              p(c_13) = [1] x1 + [6]         
                              p(c_14) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        bubble#(x',Cons(x,xs)) = [2] xs + [14]                                       
                               > [2] xs + [12]                                       
                               = 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)) =  [2] xs + [8]                                  
                                                        >= [2] xs + [8]                                  
                                                        =  bsort#(x',bubble(x,xs))                       
        
                               bsort#(S(x'),Cons(x,xs)) =  [2] xs + [8]                                  
                                                        >= [2] xs + [6]                                  
                                                        =  bubble#(x,xs)                                 
        
                                       bubble#(x,Nil()) =  [8]                                           
                                                        >= [4]                                           
                                                        =  c_3()                                         
        
        bubble[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [2] xs + [12]                                 
                                                        >= [2] xs + [12]                                 
                                                        =  c_13(bubble#(x',xs))                          
        
         bubble[Ite][False][Ite]#(True(),x',Cons(x,xs)) =  [2] xs + [12]                                 
                                                        >= [2] xs + [6]                                  
                                                        =  c_14(bubble#(x,xs))                           
        
                                        bubblesort#(xs) =  [4] xs + [4]                                  
                                                        >= [2] xs + [0]                                  
                                                        =  bsort#(len(xs),xs)                            
        
                                               <(x,0()) =  [1]                                           
                                                        >= [1]                                           
                                                        =  False()                                       
        
                                            <(0(),S(y)) =  [1]                                           
                                                        >= [1]                                           
                                                        =  True()                                        
        
                                           <(S(x),S(y)) =  [1]                                           
                                                        >= [1]                                           
                                                        =  <(x,y)                                        
        
                                        bubble(x,Nil()) =  [5]                                           
                                                        >= [5]                                           
                                                        =  Cons(x,Nil())                                 
        
                                  bubble(x',Cons(x,xs)) =  [1] xs + [8]                                  
                                                        >= [1] xs + [8]                                  
                                                        =  bubble[Ite][False][Ite](<(x',x),x',Cons(x,xs))
        
         bubble[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] xs + [8]                                  
                                                        >= [1] xs + [8]                                  
                                                        =  Cons(x,bubble(x',xs))                         
        
          bubble[Ite][False][Ite](True(),x',Cons(x,xs)) =  [1] xs + [8]                                  
                                                        >= [1] xs + [8]                                  
                                                        =  Cons(x',bubble(x,xs))                         
        
** Step 6.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            bsort#(S(x'),Cons(x,xs)) -> bsort#(x',bubble(x,xs))
            bsort#(S(x'),Cons(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))
            bubblesort#(xs) -> 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_5/2,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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))