/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[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],inssort,isort} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[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],inssort,isort} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[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],inssort,isort} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          isort(y,z){y -> Cons(x,y)} =
            isort(Cons(x,y),z) ->^+ isort(y,insert(x,z))
              = C[isort(y,insert(x,z)) = isort(y,z){z -> insert(x,z)}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            inssort(xs) -> isort(xs,Nil())
            isort(Cons(x,xs),r) -> isort(xs,insert(x,r))
            isort(Nil(),r) -> Nil()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[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],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#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
          inssort#(xs) -> c_2(isort#(xs,Nil()))
          isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
          isort#(Nil(),r) -> c_4()
        Weak DPs
          <#(x,0()) -> c_5()
          <#(0(),S(y)) -> c_6()
          <#(S(x),S(y)) -> c_7(<#(x,y))
          insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
          insert[Ite]#(True(),x,r) -> c_9()
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
            inssort#(xs) -> c_2(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
            isort#(Nil(),r) -> c_4()
        - Weak DPs:
            <#(x,0()) -> c_5()
            <#(0(),S(y)) -> c_6()
            <#(S(x),S(y)) -> c_7(<#(x,y))
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
            insert[Ite]#(True(),x,r) -> c_9()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[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) -> Nil()
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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
          {4}
        by application of
          Pre({4}) = {2,3}.
        Here rules are labelled as follows:
          1: insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
          2: inssort#(xs) -> c_2(isort#(xs,Nil()))
          3: isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
          4: isort#(Nil(),r) -> c_4()
          5: <#(x,0()) -> c_5()
          6: <#(0(),S(y)) -> c_6()
          7: <#(S(x),S(y)) -> c_7(<#(x,y))
          8: insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
          9: insert[Ite]#(True(),x,r) -> c_9()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
            inssort#(xs) -> c_2(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            <#(x,0()) -> c_5()
            <#(0(),S(y)) -> c_6()
            <#(S(x),S(y)) -> c_7(<#(x,y))
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
            insert[Ite]#(True(),x,r) -> c_9()
            isort#(Nil(),r) -> c_4()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[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) -> Nil()
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
             -->_1 insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs)):7
             -->_2 <#(S(x),S(y)) -> c_7(<#(x,y)):6
             -->_1 insert[Ite]#(True(),x,r) -> c_9():8
             -->_2 <#(x,0()) -> c_5():4
          
          2:S:inssort#(xs) -> c_2(isort#(xs,Nil()))
             -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
             -->_1 isort#(Nil(),r) -> c_4():9
          
          3:S:isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Nil(),r) -> c_4():9
             -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
             -->_2 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x)):1
          
          4:W:<#(x,0()) -> c_5()
             
          
          5:W:<#(0(),S(y)) -> c_6()
             
          
          6:W:<#(S(x),S(y)) -> c_7(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_7(<#(x,y)):6
             -->_1 <#(0(),S(y)) -> c_6():5
             -->_1 <#(x,0()) -> c_5():4
          
          7:W:insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
             -->_1 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x)):1
          
          8:W:insert[Ite]#(True(),x,r) -> c_9()
             
          
          9:W:isort#(Nil(),r) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: isort#(Nil(),r) -> c_4()
          8: insert[Ite]#(True(),x,r) -> c_9()
          6: <#(S(x),S(y)) -> c_7(<#(x,y))
          4: <#(x,0()) -> c_5()
          5: <#(0(),S(y)) -> c_6()
** Step 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
            inssort#(xs) -> c_2(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[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) -> Nil()
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x))
             -->_1 insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs)):7
          
          2:S:inssort#(xs) -> c_2(isort#(xs,Nil()))
             -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
          
          3:S:isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
             -->_2 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x)):1
          
          7:W:insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
             -->_1 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r),<#(S(x),x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
            inssort#(xs) -> c_2(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[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) -> Nil()
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
          insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
          insert[Ite](True(),x,r) -> Cons(x,r)
          insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
          insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
          inssort#(xs) -> c_2(isort#(xs,Nil()))
          isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
** Step 1.b:6: RemoveHeads.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
            inssort#(xs) -> c_2(isort#(xs,Nil()))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[Ite]#,inssort#
            ,isort#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
           -->_1 insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs)):4
        
        2:S:inssort#(xs) -> c_2(isort#(xs,Nil()))
           -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
        
        3:S:isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
           -->_1 isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r)):3
           -->_2 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r)):1
        
        4:W:insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
           -->_1 insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r)):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).
        
        [(2,inssort#(xs) -> c_2(isort#(xs,Nil())))]
** Step 1.b:7: DecomposeDG.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak DPs:
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(insert#(x',xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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_3(isort#(xs,insert(x,r)),insert#(x,r))
        and a lower component
          insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
          insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(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 1.b:7.a:1: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)),insert#(x,r))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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_3(isort#(xs,insert(x,r)),insert#(x,r))
             -->_1 isort#(Cons(x,xs),r) -> c_3(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_3(isort#(xs,insert(x,r)))
*** Step 1.b:7.a:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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_3) = {1}
        
        Following symbols are considered usable:
          {<#,insert#,insert[Ite]#,inssort#,isort#}
        TcT has computed the following interpretation:
                     p(0) = [0]          
                     p(<) = [0]          
                  p(Cons) = [1] x2 + [9] 
                 p(False) = [0]          
                   p(Nil) = [0]          
                     p(S) = [1] x1 + [0] 
                  p(True) = [0]          
                p(insert) = [0]          
           p(insert[Ite]) = [0]          
               p(inssort) = [0]          
                 p(isort) = [0]          
                    p(<#) = [0]          
               p(insert#) = [0]          
          p(insert[Ite]#) = [0]          
              p(inssort#) = [0]          
                p(isort#) = [2] x1 + [9] 
                   p(c_1) = [0]          
                   p(c_2) = [4] x1 + [0] 
                   p(c_3) = [1] x1 + [10]
                   p(c_4) = [0]          
                   p(c_5) = [1]          
                   p(c_6) = [0]          
                   p(c_7) = [0]          
                   p(c_8) = [2] x1 + [0] 
                   p(c_9) = [8]          
        
        Following rules are strictly oriented:
        isort#(Cons(x,xs),r) = [2] xs + [27]              
                             > [2] xs + [19]              
                             = c_3(isort#(xs,insert(x,r)))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:7.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            isort#(Cons(x,xs),r) -> c_3(isort#(xs,insert(x,r)))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            insert(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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 1.b:7.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
        - Weak DPs:
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(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(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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_1) = {1},
          uargs(c_8) = {1}
        
        Following symbols are considered usable:
          {insert,insert[Ite],<#,insert#,insert[Ite]#,inssort#,isort#}
        TcT has computed the following interpretation:
                     p(0) = [0]                  
                     p(<) = [6] x2 + [0]         
                  p(Cons) = [1] x1 + [1] x2 + [1]
                 p(False) = [0]                  
                   p(Nil) = [0]                  
                     p(S) = [0]                  
                  p(True) = [0]                  
                p(insert) = [1] x1 + [1] x2 + [1]
           p(insert[Ite]) = [1] x2 + [1] x3 + [1]
               p(inssort) = [1] x1 + [8]         
                 p(isort) = [1] x1 + [1]         
                    p(<#) = [1] x2 + [2]         
               p(insert#) = [4] x1 + [4] x2 + [5]
          p(insert[Ite]#) = [5] x2 + [4] x3 + [1]
              p(inssort#) = [2]                  
                p(isort#) = [4] x1 + [4] x2 + [2]
                   p(c_1) = [1] x1 + [1]         
                   p(c_2) = [1] x1 + [1]         
                   p(c_3) = [1] x1 + [4] x2 + [0]
                   p(c_4) = [1]                  
                   p(c_5) = [0]                  
                   p(c_6) = [1]                  
                   p(c_7) = [1] x1 + [2]         
                   p(c_8) = [1] x1 + [0]         
                   p(c_9) = [0]                  
        
        Following rules are strictly oriented:
        insert#(S(x),r) = [4] r + [5]                        
                        > [4] r + [2]                        
                        = c_1(insert[Ite]#(<(S(x),x),S(x),r))
        
        
        Following rules are (at-least) weakly oriented:
        insert[Ite]#(False(),x',Cons(x,xs)) =  [4] x + [5] x' + [4] xs + [5]
                                            >= [4] x' + [4] xs + [5]        
                                            =  c_8(insert#(x',xs))          
        
                       isort#(Cons(x,xs),r) =  [4] r + [4] x + [4] xs + [6] 
                                            >= [4] r + [4] x + [5]          
                                            =  insert#(x,r)                 
        
                       isort#(Cons(x,xs),r) =  [4] r + [4] x + [4] xs + [6] 
                                            >= [4] r + [4] x + [4] xs + [6] 
                                            =  isort#(xs,insert(x,r))       
        
                             insert(S(x),r) =  [1] r + [1]                  
                                            >= [1] r + [1]                  
                                            =  insert[Ite](<(S(x),x),S(x),r)
        
         insert[Ite](False(),x',Cons(x,xs)) =  [1] x + [1] x' + [1] xs + [2]
                                            >= [1] x + [1] x' + [1] xs + [2]
                                            =  Cons(x,insert(x',xs))        
        
                    insert[Ite](True(),x,r) =  [1] r + [1] x + [1]          
                                            >= [1] r + [1] x + [1]          
                                            =  Cons(x,r)                    
        
*** Step 1.b:7.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            insert#(S(x),r) -> c_1(insert[Ite]#(<(S(x),x),S(x),r))
            insert[Ite]#(False(),x',Cons(x,xs)) -> c_8(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(S(x),r) -> insert[Ite](<(S(x),x),S(x),r)
            insert[Ite](False(),x',Cons(x,xs)) -> Cons(x,insert(x',xs))
            insert[Ite](True(),x,r) -> Cons(x,r)
        - Signature:
            {</2,insert/2,insert[Ite]/3,inssort/1,isort/2,<#/2,insert#/2,insert[Ite]#/3,inssort#/1,isort#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,insert#,insert[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(Omega(n^1),O(n^2))