WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,goal,notEmpty,ordered,ordered[Ite]} and constructors {0
            ,Cons,False,Nil,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          goal#(xs) -> c_1(ordered#(xs))
          notEmpty#(Cons(x,xs)) -> c_2()
          notEmpty#(Nil()) -> c_3()
          ordered#(Cons(x,Nil())) -> c_4()
          ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          ordered#(Nil()) -> c_6()
        Weak DPs
          <#(x,0()) -> c_7()
          <#(0(),S(y)) -> c_8()
          <#(S(x),S(y)) -> c_9(<#(x,y))
          ordered[Ite]#(False(),xs) -> c_10()
          ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(ordered#(xs))
            notEmpty#(Cons(x,xs)) -> c_2()
            notEmpty#(Nil()) -> c_3()
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Strict TRS:
            goal(xs) -> ordered(xs)
            notEmpty(Cons(x,xs)) -> True()
            notEmpty(Nil()) -> False()
            ordered(Cons(x,Nil())) -> True()
            ordered(Cons(x',Cons(x,xs))) -> ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))
            ordered(Nil()) -> True()
        - Weak DPs:
            <#(x,0()) -> c_7()
            <#(0(),S(y)) -> c_8()
            <#(S(x),S(y)) -> c_9(<#(x,y))
            ordered[Ite]#(False(),xs) -> c_10()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            ordered[Ite](False(),xs) -> False()
            ordered[Ite](True(),Cons(x',Cons(x,xs))) -> ordered(xs)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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)
          <#(x,0()) -> c_7()
          <#(0(),S(y)) -> c_8()
          <#(S(x),S(y)) -> c_9(<#(x,y))
          goal#(xs) -> c_1(ordered#(xs))
          notEmpty#(Cons(x,xs)) -> c_2()
          notEmpty#(Nil()) -> c_3()
          ordered#(Cons(x,Nil())) -> c_4()
          ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          ordered#(Nil()) -> c_6()
          ordered[Ite]#(False(),xs) -> c_10()
          ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(ordered#(xs))
            notEmpty#(Cons(x,xs)) -> c_2()
            notEmpty#(Nil()) -> c_3()
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            <#(x,0()) -> c_7()
            <#(0(),S(y)) -> c_8()
            <#(S(x),S(y)) -> c_9(<#(x,y))
            ordered[Ite]#(False(),xs) -> c_10()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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
          {2,3}
        by application of
          Pre({2,3}) = {}.
        Here rules are labelled as follows:
          1: goal#(xs) -> c_1(ordered#(xs))
          2: notEmpty#(Cons(x,xs)) -> c_2()
          3: notEmpty#(Nil()) -> c_3()
          4: ordered#(Cons(x,Nil())) -> c_4()
          5: ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          6: ordered#(Nil()) -> c_6()
          7: <#(x,0()) -> c_7()
          8: <#(0(),S(y)) -> c_8()
          9: <#(S(x),S(y)) -> c_9(<#(x,y))
          10: ordered[Ite]#(False(),xs) -> c_10()
          11: ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(ordered#(xs))
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            <#(x,0()) -> c_7()
            <#(0(),S(y)) -> c_8()
            <#(S(x),S(y)) -> c_9(<#(x,y))
            notEmpty#(Cons(x,xs)) -> c_2()
            notEmpty#(Nil()) -> c_3()
            ordered[Ite]#(False(),xs) -> c_10()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:goal#(xs) -> c_1(ordered#(xs))
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):3
             -->_1 ordered#(Nil()) -> c_6():4
             -->_1 ordered#(Cons(x,Nil())) -> c_4():2
          
          2:S:ordered#(Cons(x,Nil())) -> c_4()
             
          
          3:S:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
             -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):11
             -->_1 ordered[Ite]#(False(),xs) -> c_10():10
          
          4:S:ordered#(Nil()) -> c_6()
             
          
          5:W:<#(x,0()) -> c_7()
             
          
          6:W:<#(0(),S(y)) -> c_8()
             
          
          7:W:<#(S(x),S(y)) -> c_9(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_9(<#(x,y)):7
             -->_1 <#(0(),S(y)) -> c_8():6
             -->_1 <#(x,0()) -> c_7():5
          
          8:W:notEmpty#(Cons(x,xs)) -> c_2()
             
          
          9:W:notEmpty#(Nil()) -> c_3()
             
          
          10:W:ordered[Ite]#(False(),xs) -> c_10()
             
          
          11:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
             -->_1 ordered#(Nil()) -> c_6():4
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):3
             -->_1 ordered#(Cons(x,Nil())) -> c_4():2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          9: notEmpty#(Nil()) -> c_3()
          8: notEmpty#(Cons(x,xs)) -> c_2()
          7: <#(S(x),S(y)) -> c_9(<#(x,y))
          6: <#(0(),S(y)) -> c_8()
          5: <#(x,0()) -> c_7()
          10: ordered[Ite]#(False(),xs) -> c_10()
* Step 5: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            goal#(xs) -> c_1(ordered#(xs))
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:goal#(xs) -> c_1(ordered#(xs))
           -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):3
           -->_1 ordered#(Nil()) -> c_6():4
           -->_1 ordered#(Cons(x,Nil())) -> c_4():2
        
        2:S:ordered#(Cons(x,Nil())) -> c_4()
           
        
        3:S:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
           -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):11
        
        4:S:ordered#(Nil()) -> c_6()
           
        
        11:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
           -->_1 ordered#(Nil()) -> c_6():4
           -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):3
           -->_1 ordered#(Cons(x,Nil())) -> c_4():2
        
        
        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).
        
        [(1,goal#(xs) -> c_1(ordered#(xs)))]
* Step 6: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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:
              ordered#(Cons(x,Nil())) -> c_4()
          - Weak DPs:
              ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
              ordered#(Nil()) -> c_6()
              ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
          - Signature:
              {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1
              ,ordered[Ite]#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
              ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
        
        Problem (S)
          - Strict DPs:
              ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
              ordered#(Nil()) -> c_6()
          - Weak DPs:
              ordered#(Cons(x,Nil())) -> c_4()
              ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
          - Weak TRS:
              <(x,0()) -> False()
              <(0(),S(y)) -> True()
              <(S(x),S(y)) -> <(x,y)
          - Signature:
              {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1
              ,ordered[Ite]#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0
              ,c_9/1,c_10/0,c_11/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
              ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x,Nil())) -> c_4()
        - Weak DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          2:S:ordered#(Cons(x,Nil())) -> c_4()
             
          
          3:W:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
             -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):11
          
          4:W:ordered#(Nil()) -> c_6()
             
          
          11:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
             -->_1 ordered#(Cons(x,Nil())) -> c_4():2
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):3
             -->_1 ordered#(Nil()) -> c_6():4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: ordered#(Nil()) -> c_6()
** Step 6.a:2: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x,Nil())) -> c_4()
        - Weak DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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 = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: ordered#(Cons(x,Nil())) -> c_4()
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.a:2.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x,Nil())) -> c_4()
        - Weak DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {<#,goal#,notEmpty#,ordered#,ordered[Ite]#}
        TcT has computed the following interpretation:
                      p(0) = [0]                  
                      p(<) = [4] x1 + [0]         
                   p(Cons) = [0]                  
                  p(False) = [0]                  
                    p(Nil) = [1]                  
                      p(S) = [5]                  
                   p(True) = [0]                  
                   p(goal) = [1] x1 + [2]         
               p(notEmpty) = [2] x1 + [0]         
                p(ordered) = [2] x1 + [1]         
           p(ordered[Ite]) = [1] x1 + [1] x2 + [1]
                     p(<#) = [1] x2 + [0]         
                  p(goal#) = [0]                  
              p(notEmpty#) = [1] x1 + [8]         
               p(ordered#) = [1]                  
          p(ordered[Ite]#) = [1]                  
                    p(c_1) = [1] x1 + [1]         
                    p(c_2) = [8]                  
                    p(c_3) = [1]                  
                    p(c_4) = [0]                  
                    p(c_5) = [1] x1 + [0]         
                    p(c_6) = [1]                  
                    p(c_7) = [1]                  
                    p(c_8) = [0]                  
                    p(c_9) = [0]                  
                   p(c_10) = [1]                  
                   p(c_11) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        ordered#(Cons(x,Nil())) = [1]  
                                > [0]  
                                = c_4()
        
        
        Following rules are (at-least) weakly oriented:
                    ordered#(Cons(x',Cons(x,xs))) =  [1]                                            
                                                  >= [1]                                            
                                                  =  c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
        
        ordered[Ite]#(True(),Cons(x',Cons(x,xs))) =  [1]                                            
                                                  >= [1]                                            
                                                  =  c_11(ordered#(xs))                             
        
*** Step 6.a:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ordered#(Cons(x,Nil())) -> c_4()
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:ordered#(Cons(x,Nil())) -> c_4()
             
          
          2:W:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
             -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):3
          
          3:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs)))):2
             -->_1 ordered#(Cons(x,Nil())) -> c_4():1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          3: ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
          1: ordered#(Cons(x,Nil())) -> c_4()
*** Step 6.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            ordered#(Cons(x,Nil())) -> c_4()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
             -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):4
          
          2:S:ordered#(Nil()) -> c_6()
             
          
          3:W:ordered#(Cons(x,Nil())) -> c_4()
             
          
          4:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
             -->_1 ordered#(Cons(x,Nil())) -> c_4():3
             -->_1 ordered#(Nil()) -> c_6():2
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(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: ordered#(Cons(x,Nil())) -> c_4()
** Step 6.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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: ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          2: ordered#(Nil()) -> c_6()
          
        Consider the set of all dependency pairs
          1: ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          2: ordered#(Nil()) -> c_6()
          4: ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1,2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,4}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
*** Step 6.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
        - Weak DPs:
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_11) = {1}
        
        Following symbols are considered usable:
          {<,<#,goal#,notEmpty#,ordered#,ordered[Ite]#}
        TcT has computed the following interpretation:
                      p(0) = [0]                  
                      p(<) = [1]                  
                   p(Cons) = [1] x1 + [1] x2 + [1]
                  p(False) = [0]                  
                    p(Nil) = [0]                  
                      p(S) = [1] x1 + [0]         
                   p(True) = [1]                  
                   p(goal) = [2]                  
               p(notEmpty) = [1] x1 + [2]         
                p(ordered) = [1]                  
           p(ordered[Ite]) = [4] x1 + [1]         
                     p(<#) = [1] x1 + [1] x2 + [1]
                  p(goal#) = [2]                  
              p(notEmpty#) = [2] x1 + [0]         
               p(ordered#) = [8] x1 + [15]        
          p(ordered[Ite]#) = [4] x1 + [8] x2 + [8]
                    p(c_1) = [1] x1 + [1]         
                    p(c_2) = [1]                  
                    p(c_3) = [1]                  
                    p(c_4) = [0]                  
                    p(c_5) = [1] x1 + [0]         
                    p(c_6) = [14]                 
                    p(c_7) = [1]                  
                    p(c_8) = [1]                  
                    p(c_9) = [4]                  
                   p(c_10) = [8]                  
                   p(c_11) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        ordered#(Cons(x',Cons(x,xs))) = [8] x + [8] x' + [8] xs + [31]                 
                                      > [8] x + [8] x' + [8] xs + [28]                 
                                      = c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
        
                      ordered#(Nil()) = [15]                                           
                                      > [14]                                           
                                      = c_6()                                          
        
        
        Following rules are (at-least) weakly oriented:
        ordered[Ite]#(True(),Cons(x',Cons(x,xs))) =  [8] x + [8] x' + [8] xs + [28]
                                                  >= [8] xs + [15]                 
                                                  =  c_11(ordered#(xs))            
        
                                         <(x,0()) =  [1]                           
                                                  >= [0]                           
                                                  =  False()                       
        
                                      <(0(),S(y)) =  [1]                           
                                                  >= [1]                           
                                                  =  True()                        
        
                                     <(S(x),S(y)) =  [1]                           
                                                  >= [1]                           
                                                  =  <(x,y)                        
        
*** Step 6.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
            ordered#(Nil()) -> c_6()
            ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} and constructors {0,Cons,False,Nil,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
             -->_1 ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs)):3
          
          2:W:ordered#(Nil()) -> c_6()
             
          
          3:W:ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
             -->_1 ordered#(Nil()) -> c_6():2
             -->_1 ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(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.
          1: ordered#(Cons(x',Cons(x,xs))) -> c_5(ordered[Ite]#(<(x',x),Cons(x',Cons(x,xs))))
          3: ordered[Ite]#(True(),Cons(x',Cons(x,xs))) -> c_11(ordered#(xs))
          2: ordered#(Nil()) -> c_6()
*** Step 6.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
        - Signature:
            {</2,goal/1,notEmpty/1,ordered/1,ordered[Ite]/2,<#/2,goal#/1,notEmpty#/1,ordered#/1,ordered[Ite]#/2} / {0/0
            ,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,goal#,notEmpty#,ordered#
            ,ordered[Ite]#} 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^1))