WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> r
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,insert,insert[Ite][False][Ite],inssort
            ,isort} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          insert#(x,Nil()) -> c_1()
          insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          inssort#(xs) -> c_3(isort#(xs,Nil()))
          isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          isort#(Nil(),r) -> c_5()
        Weak DPs
          <#(x,0()) -> c_6()
          <#(0(),S(y)) -> c_7()
          <#(S(x),S(y)) -> c_8(<#(x,y))
          insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          insert[Ite][False][Ite]#(True(),x,r) -> c_10()
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            inssort#(xs) -> c_3(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
            isort#(Nil(),r) -> c_5()
        - Weak DPs:
            <#(x,0()) -> c_6()
            <#(0(),S(y)) -> c_7()
            <#(S(x),S(y)) -> c_8(<#(x,y))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            insert[Ite][False][Ite]#(True(),x,r) -> c_10()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> r
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          insert(x,Nil()) -> Cons(x,Nil())
          insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
          insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
          insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
          <#(x,0()) -> c_6()
          <#(0(),S(y)) -> c_7()
          <#(S(x),S(y)) -> c_8(<#(x,y))
          insert#(x,Nil()) -> c_1()
          insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          insert[Ite][False][Ite]#(True(),x,r) -> c_10()
          inssort#(xs) -> c_3(isort#(xs,Nil()))
          isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          isort#(Nil(),r) -> c_5()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            inssort#(xs) -> c_3(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
            isort#(Nil(),r) -> c_5()
        - Weak DPs:
            <#(x,0()) -> c_6()
            <#(0(),S(y)) -> c_7()
            <#(S(x),S(y)) -> c_8(<#(x,y))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            insert[Ite][False][Ite]#(True(),x,r) -> c_10()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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
          {5}
        by application of
          Pre({5}) = {3,4}.
        Here rules are labelled as follows:
          1: insert#(x,Nil()) -> c_1()
          2: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
          3: inssort#(xs) -> c_3(isort#(xs,Nil()))
          4: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          5: isort#(Nil(),r) -> c_5()
          6: <#(x,0()) -> c_6()
          7: <#(0(),S(y)) -> c_7()
          8: <#(S(x),S(y)) -> c_8(<#(x,y))
          9: insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          10: insert[Ite][False][Ite]#(True(),x,r) -> c_10()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            inssort#(xs) -> c_3(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            <#(x,0()) -> c_6()
            <#(0(),S(y)) -> c_7()
            <#(S(x),S(y)) -> c_8(<#(x,y))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            insert[Ite][False][Ite]#(True(),x,r) -> c_10()
            isort#(Nil(),r) -> c_5()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:insert#(x,Nil()) -> c_1()
             
          
          2:S:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):8
             -->_2 <#(S(x),S(y)) -> c_8(<#(x,y)):7
             -->_1 insert[Ite][False][Ite]#(True(),x,r) -> c_10():9
             -->_2 <#(0(),S(y)) -> c_7():6
             -->_2 <#(x,0()) -> c_6():5
          
          3:S:inssort#(xs) -> c_3(isort#(xs,Nil()))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
             -->_1 isort#(Nil(),r) -> c_5():10
          
          4:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Nil(),r) -> c_5():10
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):2
             -->_2 insert#(x,Nil()) -> c_1():1
          
          5:W:<#(x,0()) -> c_6()
             
          
          6:W:<#(0(),S(y)) -> c_7()
             
          
          7:W:<#(S(x),S(y)) -> c_8(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_8(<#(x,y)):7
             -->_1 <#(0(),S(y)) -> c_7():6
             -->_1 <#(x,0()) -> c_6():5
          
          8:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):2
             -->_1 insert#(x,Nil()) -> c_1():1
          
          9:W:insert[Ite][False][Ite]#(True(),x,r) -> c_10()
             
          
          10:W:isort#(Nil(),r) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          10: isort#(Nil(),r) -> c_5()
          9: insert[Ite][False][Ite]#(True(),x,r) -> c_10()
          7: <#(S(x),S(y)) -> c_8(<#(x,y))
          5: <#(x,0()) -> c_6()
          6: <#(0(),S(y)) -> c_7()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
            inssort#(xs) -> c_3(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:insert#(x,Nil()) -> c_1()
             
          
          2:S:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):8
          
          3:S:inssort#(xs) -> c_3(isort#(xs,Nil()))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
          
          4:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):2
             -->_2 insert#(x,Nil()) -> c_1():1
          
          8:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)),<#(x',x)):2
             -->_1 insert#(x,Nil()) -> c_1():1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
* Step 6: RemoveHeads WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            inssort#(xs) -> c_3(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:insert#(x,Nil()) -> c_1()
           
        
        2:S:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
           -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):5
        
        3:S:inssort#(xs) -> c_3(isort#(xs,Nil()))
           -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
        
        4:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
           -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
           -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
           -->_2 insert#(x,Nil()) -> c_1():1
        
        5:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
           -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
           -->_1 insert#(x,Nil()) -> c_1():1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(3,inssort#(xs) -> c_3(isort#(xs,Nil())))]
* Step 7: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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:
              insert#(x,Nil()) -> c_1()
          - Weak DPs:
              insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
              insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
              isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              insert(x,Nil()) -> Cons(x,Nil())
              insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
              insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
          - Signature:
              {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
              ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0
              ,c_8/1,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
              ,isort#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
              isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          - Weak DPs:
              insert#(x,Nil()) -> c_1()
              insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              insert(x,Nil()) -> Cons(x,Nil())
              insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
              insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
          - Signature:
              {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
              ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0
              ,c_8/1,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
              ,isort#} and constructors {0,Cons,False,Nil,S,True}
** Step 7.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
        - Weak DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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: insert#(x,Nil()) -> c_1()
          
        The strictly oriented rules are moved into the weak component.
*** Step 7.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            insert#(x,Nil()) -> c_1()
        - Weak DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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},
          uargs(c_4) = {1,2},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {<,<#,insert#,insert[Ite][False][Ite]#,inssort#,isort#}
        TcT has computed the following interpretation:
                                 p(0) = [0]                  
                                 p(<) = [1]                  
                              p(Cons) = [1] x2 + [2]         
                             p(False) = [1]                  
                               p(Nil) = [0]                  
                                 p(S) = [4]                  
                              p(True) = [0]                  
                            p(insert) = [0]                  
           p(insert[Ite][False][Ite]) = [2] x2 + [10]        
                           p(inssort) = [2] x1 + [1]         
                             p(isort) = [4] x1 + [1] x2 + [1]
                                p(<#) = [1] x1 + [1] x2 + [1]
                           p(insert#) = [2]                  
          p(insert[Ite][False][Ite]#) = [2] x1 + [0]         
                          p(inssort#) = [1]                  
                            p(isort#) = [9] x1 + [1]         
                               p(c_1) = [0]                  
                               p(c_2) = [1] x1 + [0]         
                               p(c_3) = [1] x1 + [8]         
                               p(c_4) = [1] x1 + [8] x2 + [2]
                               p(c_5) = [2]                  
                               p(c_6) = [4]                  
                               p(c_7) = [1]                  
                               p(c_8) = [1] x1 + [0]         
                               p(c_9) = [1] x1 + [0]         
                              p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        insert#(x,Nil()) = [2]  
                         > [0]  
                         = c_1()
        
        
        Following rules are (at-least) weakly oriented:
                                 insert#(x',Cons(x,xs)) =  [2]                                                 
                                                        >= [2]                                                 
                                                        =  c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [2]                                                 
                                                        >= [2]                                                 
                                                        =  c_9(insert#(x',xs))                                 
        
                                   isort#(Cons(x,xs),r) =  [9] xs + [19]                                       
                                                        >= [9] xs + [19]                                       
                                                        =  c_4(isort#(xs,insert(x,r)),insert#(x,r))            
        
                                               <(x,0()) =  [1]                                                 
                                                        >= [1]                                                 
                                                        =  False()                                             
        
                                            <(0(),S(y)) =  [1]                                                 
                                                        >= [0]                                                 
                                                        =  True()                                              
        
                                           <(S(x),S(y)) =  [1]                                                 
                                                        >= [1]                                                 
                                                        =  <(x,y)                                              
        
*** Step 7.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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 7.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            insert#(x,Nil()) -> c_1()
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:insert#(x,Nil()) -> c_1()
             
          
          2:W:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):3
          
          3:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
             -->_1 insert#(x,Nil()) -> c_1():1
          
          4:W:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):4
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
             -->_2 insert#(x,Nil()) -> c_1():1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          2: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          3: insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          1: insert#(x,Nil()) -> c_1()
*** Step 7.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert#(x,Nil()) -> c_1()
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):4
          
          2:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_2 insert#(x,Nil()) -> c_1():3
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):2
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
          3:W:insert#(x,Nil()) -> c_1()
             
          
          4:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x,Nil()) -> c_1():3
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: insert#(x,Nil()) -> c_1()
** Step 7.b:2: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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:
              insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          - Weak DPs:
              insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
              isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              insert(x,Nil()) -> Cons(x,Nil())
              insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
              insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
          - Signature:
              {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
              ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0
              ,c_8/1,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
              ,isort#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          - Weak DPs:
              insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
              insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
              insert(x,Nil()) -> Cons(x,Nil())
              insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
              insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
              insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
          - Signature:
              {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
              ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0
              ,c_8/1,c_9/1,c_10/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
              ,isort#} and constructors {0,Cons,False,Nil,S,True}
*** Step 7.b:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          
        Consider the set of all dependency pairs
          1: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          2: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          4: insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',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}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 7.b:2.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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},
          uargs(c_4) = {1,2},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {insert,insert[Ite][False][Ite],<#,insert#,insert[Ite][False][Ite]#,inssort#,isort#}
        TcT has computed the following interpretation:
                                 p(0) = 0                       
                                 p(<) = x1*x2 + 2*x1^2          
                              p(Cons) = 1 + x2                  
                             p(False) = 0                       
                               p(Nil) = 0                       
                                 p(S) = x1                      
                              p(True) = 0                       
                            p(insert) = 2 + x2                  
           p(insert[Ite][False][Ite]) = 2 + x3                  
                           p(inssort) = 0                       
                             p(isort) = 1 + x1^2 + x2           
                                p(<#) = 1 + 2*x1 + 2*x2 + 2*x2^2
                           p(insert#) = 3 + x2                  
          p(insert[Ite][False][Ite]#) = 2 + x3                  
                          p(inssort#) = 0                       
                            p(isort#) = 3*x1 + x1*x2 + 2*x1^2   
                               p(c_1) = 0                       
                               p(c_2) = x1                      
                               p(c_3) = x1                      
                               p(c_4) = 2 + x1 + x2             
                               p(c_5) = 0                       
                               p(c_6) = 0                       
                               p(c_7) = 0                       
                               p(c_8) = x1                      
                               p(c_9) = x1                      
                              p(c_10) = 0                       
        
        Following rules are strictly oriented:
        insert#(x',Cons(x,xs)) = 4 + xs                                              
                               > 3 + xs                                              
                               = c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
        insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  3 + xs                                        
                                                        >= 3 + xs                                        
                                                        =  c_9(insert#(x',xs))                           
        
                                   isort#(Cons(x,xs),r) =  5 + r + r*xs + 7*xs + 2*xs^2                  
                                                        >= 5 + r + r*xs + 5*xs + 2*xs^2                  
                                                        =  c_4(isort#(xs,insert(x,r)),insert#(x,r))      
        
                                        insert(x,Nil()) =  2                                             
                                                        >= 1                                             
                                                        =  Cons(x,Nil())                                 
        
                                  insert(x',Cons(x,xs)) =  3 + xs                                        
                                                        >= 3 + xs                                        
                                                        =  insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
        
         insert[Ite][False][Ite](False(),x',Cons(x,xs)) =  3 + xs                                        
                                                        >= 3 + xs                                        
                                                        =  Cons(x,insert(x',xs))                         
        
                    insert[Ite][False][Ite](True(),x,r) =  2 + r                                         
                                                        >= 1 + r                                         
                                                        =  Cons(x,r)                                     
        
**** Step 7.b:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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 7.b:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):2
          
          2:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
          3:W:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):3
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
          1: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          2: insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
**** Step 7.b:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 7.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_2 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):1
          
          2:W:insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
             -->_1 insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs)):3
          
          3:W:insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
             -->_1 insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          3: insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
*** Step 7.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
*** Step 7.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
          
        The strictly oriented rules are moved into the weak component.
**** Step 7.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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_4) = {1}
        
        Following symbols are considered usable:
          {<#,insert#,insert[Ite][False][Ite]#,inssort#,isort#}
        TcT has computed the following interpretation:
                                 p(0) = [3]                  
                                 p(<) = [5] x1 + [7] x2 + [0]
                              p(Cons) = [1] x1 + [1] x2 + [4]
                             p(False) = [1]                  
                               p(Nil) = [10]                 
                                 p(S) = [1]                  
                              p(True) = [0]                  
                            p(insert) = [8] x1 + [0]         
           p(insert[Ite][False][Ite]) = [0]                  
                           p(inssort) = [0]                  
                             p(isort) = [0]                  
                                p(<#) = [1] x2 + [0]         
                           p(insert#) = [1] x1 + [0]         
          p(insert[Ite][False][Ite]#) = [8] x2 + [8] x3 + [0]
                          p(inssort#) = [1] x1 + [0]         
                            p(isort#) = [4] x1 + [9]         
                               p(c_1) = [0]                  
                               p(c_2) = [0]                  
                               p(c_3) = [0]                  
                               p(c_4) = [1] x1 + [5]         
                               p(c_5) = [0]                  
                               p(c_6) = [0]                  
                               p(c_7) = [0]                  
                               p(c_8) = [1] x1 + [1]         
                               p(c_9) = [2] x1 + [0]         
                              p(c_10) = [1]                  
        
        Following rules are strictly oriented:
        isort#(Cons(x,xs),r) = [4] x + [4] xs + [25]      
                             > [4] xs + [14]              
                             = c_4(isort#(xs,insert(x,r)))
        
        
        Following rules are (at-least) weakly oriented:
        
**** Step 7.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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 7.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
             -->_1 isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)))
**** Step 7.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(x,Nil()) -> Cons(x,Nil())
            insert(x',Cons(x,xs)) -> insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
            insert[Ite][False][Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite][False][Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite][False][Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite][False][Ite]#/3
            ,inssort#/1,isort#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1
            ,c_9/1,c_10/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite][False][Ite]#,inssort#
            ,isort#} 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))