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: 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)
            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:
        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 3: 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)
            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:
        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 4: 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)
            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:
        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 5: 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)))
            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)
            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_2/1,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:
        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)
          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))
          inssort#(xs) -> c_3(isort#(xs,Nil()))
          isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
* 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: DecomposeDG 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:
        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
          isort#(Cons(x,xs),r) -> c_4(isort#(xs,insert(x,r)),insert#(x,r))
        and a lower component
          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))
        Further, following extension rules are added to the lower component.
          isort#(Cons(x,xs),r) -> insert#(x,r)
          isort#(Cons(x,xs),r) -> isort#(xs,insert(x,r))
** Step 7.a:1: 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.a:2: WeightGap 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:
        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(Cons) = {2},
            uargs(insert[Ite][False][Ite]) = {1},
            uargs(isort#) = {2},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                                   p(0) = [0]                           
                                   p(<) = [0]                           
                                p(Cons) = [1] x1 + [1] x2 + [2]         
                               p(False) = [0]                           
                                 p(Nil) = [0]                           
                                   p(S) = [1]                           
                                p(True) = [0]                           
                              p(insert) = [1] x1 + [1] x2 + [2]         
             p(insert[Ite][False][Ite]) = [1] x1 + [1] x2 + [1] x3 + [2]
                             p(inssort) = [0]                           
                               p(isort) = [1] x1 + [1] x2 + [1]         
                                  p(<#) = [1] x1 + [1] x2 + [0]         
                             p(insert#) = [1] x1 + [2]                  
            p(insert[Ite][False][Ite]#) = [4] x1 + [2] x2 + [1] x3 + [1]
                            p(inssort#) = [1] x1 + [1]                  
                              p(isort#) = [5] x1 + [1] x2 + [1]         
                                 p(c_1) = [0]                           
                                 p(c_2) = [1] x1 + [1]                  
                                 p(c_3) = [1] x1 + [1]                  
                                 p(c_4) = [1] x1 + [1]                  
                                 p(c_5) = [0]                           
                                 p(c_6) = [1]                           
                                 p(c_7) = [0]                           
                                 p(c_8) = [0]                           
                                 p(c_9) = [0]                           
                                p(c_10) = [2]                           
          
          Following rules are strictly oriented:
          isort#(Cons(x,xs),r) = [1] r + [5] x + [5] xs + [11]
                               > [1] r + [1] x + [5] xs + [4] 
                               = c_4(isort#(xs,insert(x,r)))  
          
          
          Following rules are (at-least) weakly oriented:
                                                <(x,0()) =  [0]                                           
                                                         >= [0]                                           
                                                         =  False()                                       
          
                                             <(0(),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  True()                                        
          
                                            <(S(x),S(y)) =  [0]                                           
                                                         >= [0]                                           
                                                         =  <(x,y)                                        
          
                                         insert(x,Nil()) =  [1] x + [2]                                   
                                                         >= [1] x + [2]                                   
                                                         =  Cons(x,Nil())                                 
          
                                   insert(x',Cons(x,xs)) =  [1] x + [1] x' + [1] xs + [4]                 
                                                         >= [1] x + [1] x' + [1] xs + [4]                 
                                                         =  insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
          
          insert[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] x + [1] x' + [1] xs + [4]                 
                                                         >= [1] x + [1] x' + [1] xs + [4]                 
                                                         =  Cons(x,insert(x',xs))                         
          
                     insert[Ite][False][Ite](True(),x,r) =  [1] r + [1] x + [2]                           
                                                         >= [1] r + [1] x + [2]                           
                                                         =  Cons(x,r)                                     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 7.a:3: EmptyProcessor 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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 7.b:1: WeightGap WORST_CASE(?,O(n^1))
    + 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)))
        - Weak DPs:
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> insert#(x,r)
            isort#(Cons(x,xs),r) -> 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_4/2,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:
        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(Cons) = {2},
            uargs(insert[Ite][False][Ite]) = {1},
            uargs(insert[Ite][False][Ite]#) = {1},
            uargs(isort#) = {2},
            uargs(c_2) = {1},
            uargs(c_9) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                                   p(0) = [1]                  
                                   p(<) = [0]                  
                                p(Cons) = [1] x2 + [0]         
                               p(False) = [0]                  
                                 p(Nil) = [3]                  
                                   p(S) = [0]                  
                                p(True) = [0]                  
                              p(insert) = [1] x2 + [0]         
             p(insert[Ite][False][Ite]) = [1] x1 + [1] x3 + [0]
                             p(inssort) = [1] x1 + [1]         
                               p(isort) = [1] x1 + [0]         
                                  p(<#) = [2]                  
                             p(insert#) = [1]                  
            p(insert[Ite][False][Ite]#) = [1] x1 + [1]         
                            p(inssort#) = [1] x1 + [4]         
                              p(isort#) = [1] x2 + [3]         
                                 p(c_1) = [0]                  
                                 p(c_2) = [1] x1 + [6]         
                                 p(c_3) = [0]                  
                                 p(c_4) = [4] x1 + [1] x2 + [0]
                                 p(c_5) = [0]                  
                                 p(c_6) = [4]                  
                                 p(c_7) = [2]                  
                                 p(c_8) = [4] x1 + [1]         
                                 p(c_9) = [1] x1 + [0]         
                                p(c_10) = [0]                  
          
          Following rules are strictly oriented:
          insert#(x,Nil()) = [1]  
                           > [0]  
                           = c_1()
          
          
          Following rules are (at-least) weakly oriented:
                                   insert#(x',Cons(x,xs)) =  [1]                                                 
                                                          >= [7]                                                 
                                                          =  c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
          
          insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [1]                                                 
                                                          >= [1]                                                 
                                                          =  c_9(insert#(x',xs))                                 
          
                                     isort#(Cons(x,xs),r) =  [1] r + [3]                                         
                                                          >= [1]                                                 
                                                          =  insert#(x,r)                                        
          
                                     isort#(Cons(x,xs),r) =  [1] r + [3]                                         
                                                          >= [1] r + [3]                                         
                                                          =  isort#(xs,insert(x,r))                              
          
                                                 <(x,0()) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  False()                                             
          
                                              <(0(),S(y)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  True()                                              
          
                                             <(S(x),S(y)) =  [0]                                                 
                                                          >= [0]                                                 
                                                          =  <(x,y)                                              
          
                                          insert(x,Nil()) =  [3]                                                 
                                                          >= [3]                                                 
                                                          =  Cons(x,Nil())                                       
          
                                    insert(x',Cons(x,xs)) =  [1] xs + [0]                                        
                                                          >= [1] xs + [0]                                        
                                                          =  insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))      
          
           insert[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] xs + [0]                                        
                                                          >= [1] xs + [0]                                        
                                                          =  Cons(x,insert(x',xs))                               
          
                      insert[Ite][False][Ite](True(),x,r) =  [1] r + [0]                                         
                                                          >= [1] r + [0]                                         
                                                          =  Cons(x,r)                                           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 7.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            insert#(x',Cons(x,xs)) -> c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        - Weak DPs:
            insert#(x,Nil()) -> c_1()
            insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) -> c_9(insert#(x',xs))
            isort#(Cons(x,xs),r) -> insert#(x,r)
            isort#(Cons(x,xs),r) -> 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_4/2,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 any 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_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) = [1]                  
                                 p(<) = [0]                  
                              p(Cons) = [1] x1 + [1] x2 + [4]
                             p(False) = [0]                  
                               p(Nil) = [0]                  
                                 p(S) = [1] x1 + [1]         
                              p(True) = [1]                  
                            p(insert) = [1] x1 + [1] x2 + [4]
           p(insert[Ite][False][Ite]) = [1] x2 + [1] x3 + [4]
                           p(inssort) = [1] x1 + [0]         
                             p(isort) = [1] x1 + [2]         
                                p(<#) = [1] x2 + [1]         
                           p(insert#) = [1] x1 + [4] x2 + [8]
          p(insert[Ite][False][Ite]#) = [1] x2 + [4] x3 + [1]
                          p(inssort#) = [2]                  
                            p(isort#) = [5] x1 + [4] x2 + [8]
                               p(c_1) = [0]                  
                               p(c_2) = [1] x1 + [6]         
                               p(c_3) = [1] x1 + [1]         
                               p(c_4) = [1] x2 + [4]         
                               p(c_5) = [1]                  
                               p(c_6) = [1]                  
                               p(c_7) = [0]                  
                               p(c_8) = [1] x1 + [2]         
                               p(c_9) = [1] x1 + [9]         
                              p(c_10) = [0]                  
        
        Following rules are strictly oriented:
        insert#(x',Cons(x,xs)) = [4] x + [1] x' + [4] xs + [24]                      
                               > [4] x + [1] x' + [4] xs + [23]                      
                               = c_2(insert[Ite][False][Ite]#(<(x',x),x',Cons(x,xs)))
        
        
        Following rules are (at-least) weakly oriented:
                                       insert#(x,Nil()) =  [1] x + [8]                                   
                                                        >= [0]                                           
                                                        =  c_1()                                         
        
        insert[Ite][False][Ite]#(False(),x',Cons(x,xs)) =  [4] x + [1] x' + [4] xs + [17]                
                                                        >= [1] x' + [4] xs + [17]                        
                                                        =  c_9(insert#(x',xs))                           
        
                                   isort#(Cons(x,xs),r) =  [4] r + [5] x + [5] xs + [28]                 
                                                        >= [4] r + [1] x + [8]                           
                                                        =  insert#(x,r)                                  
        
                                   isort#(Cons(x,xs),r) =  [4] r + [5] x + [5] xs + [28]                 
                                                        >= [4] r + [4] x + [5] xs + [24]                 
                                                        =  isort#(xs,insert(x,r))                        
        
                                        insert(x,Nil()) =  [1] x + [4]                                   
                                                        >= [1] x + [4]                                   
                                                        =  Cons(x,Nil())                                 
        
                                  insert(x',Cons(x,xs)) =  [1] x + [1] x' + [1] xs + [8]                 
                                                        >= [1] x + [1] x' + [1] xs + [8]                 
                                                        =  insert[Ite][False][Ite](<(x',x),x',Cons(x,xs))
        
         insert[Ite][False][Ite](False(),x',Cons(x,xs)) =  [1] x + [1] x' + [1] xs + [8]                 
                                                        >= [1] x + [1] x' + [1] xs + [8]                 
                                                        =  Cons(x,insert(x',xs))                         
        
                    insert[Ite][False][Ite](True(),x,r) =  [1] r + [1] x + [4]                           
                                                        >= [1] r + [1] x + [4]                           
                                                        =  Cons(x,r)                                     
        
** Step 7.b:3: EmptyProcessor 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) -> insert#(x,r)
            isort#(Cons(x,xs),r) -> 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_4/2,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).

WORST_CASE(?,O(n^2))