* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            avg(xs) -> quot(hd(sum(xs)),length(xs))
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(-(x,y),s(y)))
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1} / {0/0,:/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,++,-,avg,hd,length,quot,sum} and constructors {0,:,nil
            ,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            avg(xs) -> quot(hd(sum(xs)),length(xs))
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(-(x,y),s(y)))
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1} / {0/0,:/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,++,-,avg,hd,length,quot,sum} and constructors {0,:,nil
            ,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){x -> s(x)} =
            +(s(x),y) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            avg(xs) -> quot(hd(sum(xs)),length(xs))
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(-(x,y),s(y)))
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1} / {0/0,:/2,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,++,-,avg,hd,length,quot,sum} and constructors {0,:,nil
            ,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(0(),y) -> c_1()
          +#(s(x),y) -> c_2(+#(x,y))
          ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          ++#(nil(),ys) -> c_4()
          -#(x,0()) -> c_5()
          -#(0(),s(y)) -> c_6()
          -#(s(x),s(y)) -> c_7(-#(x,y))
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
          hd#(:(x,xs)) -> c_9()
          length#(:(x,xs)) -> c_10(length#(xs))
          length#(nil()) -> c_11()
          quot#(0(),s(y)) -> c_12()
          quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                           ,++#(xs,sum(:(x,:(y,ys))))
                                           ,sum#(:(x,:(y,ys))))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          sum#(:(x,nil())) -> c_16()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(0(),y) -> c_1()
            +#(s(x),y) -> c_2(+#(x,y))
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            ++#(nil(),ys) -> c_4()
            -#(x,0()) -> c_5()
            -#(0(),s(y)) -> c_6()
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
            hd#(:(x,xs)) -> c_9()
            length#(:(x,xs)) -> c_10(length#(xs))
            length#(nil()) -> c_11()
            quot#(0(),s(y)) -> c_12()
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
            sum#(:(x,nil())) -> c_16()
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            avg(xs) -> quot(hd(sum(xs)),length(xs))
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(-(x,y),s(y)))
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          ++(:(x,xs),ys) -> :(x,++(xs,ys))
          ++(nil(),ys) -> ys
          -(x,0()) -> x
          -(0(),s(y)) -> 0()
          -(s(x),s(y)) -> -(x,y)
          hd(:(x,xs)) -> x
          length(:(x,xs)) -> s(length(xs))
          length(nil()) -> 0()
          sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
          sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
          sum(:(x,nil())) -> :(x,nil())
          +#(0(),y) -> c_1()
          +#(s(x),y) -> c_2(+#(x,y))
          ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          ++#(nil(),ys) -> c_4()
          -#(x,0()) -> c_5()
          -#(0(),s(y)) -> c_6()
          -#(s(x),s(y)) -> c_7(-#(x,y))
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
          hd#(:(x,xs)) -> c_9()
          length#(:(x,xs)) -> c_10(length#(xs))
          length#(nil()) -> c_11()
          quot#(0(),s(y)) -> c_12()
          quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                           ,++#(xs,sum(:(x,:(y,ys))))
                                           ,sum#(:(x,:(y,ys))))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          sum#(:(x,nil())) -> c_16()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(0(),y) -> c_1()
            +#(s(x),y) -> c_2(+#(x,y))
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            ++#(nil(),ys) -> c_4()
            -#(x,0()) -> c_5()
            -#(0(),s(y)) -> c_6()
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
            hd#(:(x,xs)) -> c_9()
            length#(:(x,xs)) -> c_10(length#(xs))
            length#(nil()) -> c_11()
            quot#(0(),s(y)) -> c_12()
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
            sum#(:(x,nil())) -> c_16()
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,5,6,9,11,12,16}
        by application of
          Pre({1,4,5,6,9,11,12,16}) = {2,3,7,8,10,13,14,15}.
        Here rules are labelled as follows:
          1: +#(0(),y) -> c_1()
          2: +#(s(x),y) -> c_2(+#(x,y))
          3: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          4: ++#(nil(),ys) -> c_4()
          5: -#(x,0()) -> c_5()
          6: -#(0(),s(y)) -> c_6()
          7: -#(s(x),s(y)) -> c_7(-#(x,y))
          8: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
          9: hd#(:(x,xs)) -> c_9()
          10: length#(:(x,xs)) -> c_10(length#(xs))
          11: length#(nil()) -> c_11()
          12: quot#(0(),s(y)) -> c_12()
          13: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          14: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
          15: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          16: sum#(:(x,nil())) -> c_16()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak DPs:
            +#(0(),y) -> c_1()
            ++#(nil(),ys) -> c_4()
            -#(x,0()) -> c_5()
            -#(0(),s(y)) -> c_6()
            hd#(:(x,xs)) -> c_9()
            length#(nil()) -> c_11()
            quot#(0(),s(y)) -> c_12()
            sum#(:(x,nil())) -> c_16()
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(0(),y) -> c_1():9
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          2:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(nil(),ys) -> c_4():10
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
          
          3:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(0(),s(y)) -> c_6():12
             -->_1 -#(x,0()) -> c_5():11
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3
          
          4:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_3 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
             -->_4 length#(:(x,xs)) -> c_10(length#(xs)):5
             -->_3 sum#(:(x,nil())) -> c_16():16
             -->_1 quot#(0(),s(y)) -> c_12():15
             -->_4 length#(nil()) -> c_11():14
             -->_2 hd#(:(x,xs)) -> c_9():13
          
          5:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(nil()) -> c_11():14
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5
          
          6:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(0(),s(y)) -> c_12():15
             -->_2 -#(0(),s(y)) -> c_6():12
             -->_2 -#(x,0()) -> c_5():11
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3
          
          7:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(:(x,nil())) -> c_16():16
             -->_2 ++#(nil(),ys) -> c_4():10
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
          
          8:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_1 sum#(:(x,nil())) -> c_16():16
             -->_2 +#(0(),y) -> c_1():9
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_2 +#(s(x),y) -> c_2(+#(x,y)):1
          
          9:W:+#(0(),y) -> c_1()
             
          
          10:W:++#(nil(),ys) -> c_4()
             
          
          11:W:-#(x,0()) -> c_5()
             
          
          12:W:-#(0(),s(y)) -> c_6()
             
          
          13:W:hd#(:(x,xs)) -> c_9()
             
          
          14:W:length#(nil()) -> c_11()
             
          
          15:W:quot#(0(),s(y)) -> c_12()
             
          
          16:W:sum#(:(x,nil())) -> c_16()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          13: hd#(:(x,xs)) -> c_9()
          14: length#(nil()) -> c_11()
          15: quot#(0(),s(y)) -> c_12()
          16: sum#(:(x,nil())) -> c_16()
          11: -#(x,0()) -> c_5()
          12: -#(0(),s(y)) -> c_6()
          10: ++#(nil(),ys) -> c_4()
          9: +#(0(),y) -> c_1()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          2:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
          
          3:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3
          
          4:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_3 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
             -->_4 length#(:(x,xs)) -> c_10(length#(xs)):5
          
          5:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5
          
          6:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3
          
          7:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
          
          8:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_2 +#(s(x),y) -> c_2(+#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
** Step 1.b:6: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              +#(s(x),y) -> c_2(+#(x,y))
          - Weak DPs:
              ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
              -#(s(x),s(y)) -> c_7(-#(x,y))
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/3,c_15/2,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
              -#(s(x),s(y)) -> c_7(-#(x,y))
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          - Weak DPs:
              +#(s(x),y) -> c_2(+#(x,y))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/3,c_15/2,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
*** Step 1.b:6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          2:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
          
          3:W:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):5
          
          5:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5
          
          6:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6
          
          7:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
          
          8:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_2 +#(s(x),y) -> c_2(+#(x,y)):1
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: length#(:(x,xs)) -> c_10(length#(xs))
          6: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          3: -#(s(x),s(y)) -> c_7(-#(x,y))
          2: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
*** Step 1.b:6.a:2: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
          
          7:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):7
          
          8:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_2 +#(s(x),y) -> c_2(+#(x,y)):1
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
*** Step 1.b:6.a:3: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          ++(:(x,xs),ys) -> :(x,++(xs,ys))
          ++(nil(),ys) -> ys
          sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
          sum(:(x,nil())) -> :(x,nil())
          +#(s(x),y) -> c_2(+#(x,y))
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
*** Step 1.b:6.a:4: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        and a lower component
          +#(s(x),y) -> c_2(+#(x,y))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        Further, following extension rules are added to the lower component.
          avg#(xs) -> sum#(xs)
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
**** Step 1.b:6.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          
        Consider the set of all dependency pairs
          1: avg#(xs) -> c_8(sum#(xs))
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        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
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
***** Step 1.b:6.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [8] x1 + [0]         
               p(++) = [6] x1 + [2] x2 + [2]
                p(-) = [1] x1 + [2] x2 + [0]
                p(0) = [0]                  
                p(:) = [1] x2 + [4]         
              p(avg) = [1] x1 + [1]         
               p(hd) = [4] x1 + [0]         
           p(length) = [0]                  
              p(nil) = [3]                  
             p(quot) = [1] x1 + [8] x2 + [1]
                p(s) = [1] x1 + [2]         
              p(sum) = [7]                  
               p(+#) = [1] x1 + [0]         
              p(++#) = [1]                  
               p(-#) = [1] x1 + [1] x2 + [1]
             p(avg#) = [2] x1 + [2]         
              p(hd#) = [2] x1 + [0]         
          p(length#) = [1]                  
            p(quot#) = [1] x1 + [1] x2 + [2]
             p(sum#) = [1] x1 + [1]         
              p(c_1) = [1]                  
              p(c_2) = [0]                  
              p(c_3) = [2] x1 + [0]         
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [1]                  
              p(c_7) = [1] x1 + [1]         
              p(c_8) = [2] x1 + [0]         
              p(c_9) = [1]                  
             p(c_10) = [2]                  
             p(c_11) = [1]                  
             p(c_12) = [1]                  
             p(c_13) = [2]                  
             p(c_14) = [1] x1 + [0]         
             p(c_15) = [2] x1 + [1] x2 + [0]
             p(c_16) = [0]                  
        
        Following rules are strictly oriented:
        sum#(++(xs,:(x,:(y,ys)))) = [6] xs + [2] ys + [19]                                 
                                  > [6] xs + [17]                                          
                                  = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        
        
        Following rules are (at-least) weakly oriented:
                 avg#(xs) =  [2] xs + [2]          
                          >= [2] xs + [2]          
                          =  c_8(sum#(xs))         
        
           ++(:(x,xs),ys) =  [6] xs + [2] ys + [26]
                          >= [6] xs + [2] ys + [6] 
                          =  :(x,++(xs,ys))        
        
             ++(nil(),ys) =  [2] ys + [20]         
                          >= [1] ys + [0]          
                          =  ys                    
        
        sum(:(x,:(y,xs))) =  [7]                   
                          >= [7]                   
                          =  sum(:(+(x,y),xs))     
        
          sum(:(x,nil())) =  [7]                   
                          >= [7]                   
                          =  :(x,nil())            
        
***** Step 1.b:6.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 1.b:6.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(sum#(xs))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(sum#(xs))
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
***** Step 1.b:6.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.a:4.b:1: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          avg#(xs) -> sum#(xs)
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        and a lower component
          +#(s(x),y) -> c_2(+#(x,y))
        Further, following extension rules are added to the lower component.
          avg#(xs) -> sum#(xs)
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
          sum#(:(x,:(y,xs))) -> +#(x,y)
          sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
***** Step 1.b:6.a:4.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          
        Consider the set of all dependency pairs
          1: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
          2: avg#(xs) -> sum#(xs)
          3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
        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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.a:4.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_15) = {1}
        
        Following symbols are considered usable:
          {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [1] x2 + [1]         
               p(++) = [1] x1 + [1] x2 + [0]
                p(-) = [1] x1 + [2] x2 + [0]
                p(0) = [4]                  
                p(:) = [1] x1 + [1] x2 + [3]
              p(avg) = [1]                  
               p(hd) = [0]                  
           p(length) = [0]                  
              p(nil) = [2]                  
             p(quot) = [1] x1 + [0]         
                p(s) = [1] x1 + [0]         
              p(sum) = [1] x1 + [0]         
               p(+#) = [0]                  
              p(++#) = [1] x2 + [2]         
               p(-#) = [2] x1 + [1]         
             p(avg#) = [5] x1 + [0]         
              p(hd#) = [1] x1 + [4]         
          p(length#) = [0]                  
            p(quot#) = [2] x2 + [2]         
             p(sum#) = [4] x1 + [0]         
              p(c_1) = [0]                  
              p(c_2) = [2] x1 + [0]         
              p(c_3) = [1] x1 + [1]         
              p(c_4) = [2]                  
              p(c_5) = [2]                  
              p(c_6) = [1]                  
              p(c_7) = [2] x1 + [0]         
              p(c_8) = [1] x1 + [0]         
              p(c_9) = [0]                  
             p(c_10) = [1]                  
             p(c_11) = [2]                  
             p(c_12) = [0]                  
             p(c_13) = [1] x1 + [1] x2 + [0]
             p(c_14) = [1] x1 + [0]         
             p(c_15) = [1] x1 + [8] x2 + [0]
             p(c_16) = [1]                  
        
        Following rules are strictly oriented:
        sum#(:(x,:(y,xs))) = [4] x + [4] xs + [4] y + [24]   
                           > [4] xs + [4] y + [16]           
                           = c_15(sum#(:(+(x,y),xs)),+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
                         avg#(xs) =  [5] xs + [0]                          
                                  >= [4] xs + [0]                          
                                  =  sum#(xs)                              
        
        sum#(++(xs,:(x,:(y,ys)))) =  [4] x + [4] xs + [4] y + [4] ys + [24]
                                  >= [4] x + [4] xs + [4] y + [4] ys + [24]
                                  =  sum#(++(xs,sum(:(x,:(y,ys)))))        
        
        sum#(++(xs,:(x,:(y,ys)))) =  [4] x + [4] xs + [4] y + [4] ys + [24]
                                  >= [4] x + [4] y + [4] ys + [24]         
                                  =  sum#(:(x,:(y,ys)))                    
        
                         +(0(),y) =  [1] y + [1]                           
                                  >= [1] y + [0]                           
                                  =  y                                     
        
                        +(s(x),y) =  [1] y + [1]                           
                                  >= [1] y + [1]                           
                                  =  s(+(x,y))                             
        
                   ++(:(x,xs),ys) =  [1] x + [1] xs + [1] ys + [3]         
                                  >= [1] x + [1] xs + [1] ys + [3]         
                                  =  :(x,++(xs,ys))                        
        
                     ++(nil(),ys) =  [1] ys + [2]                          
                                  >= [1] ys + [0]                          
                                  =  ys                                    
        
                sum(:(x,:(y,xs))) =  [1] x + [1] xs + [1] y + [6]          
                                  >= [1] xs + [1] y + [4]                  
                                  =  sum(:(+(x,y),xs))                     
        
                  sum(:(x,nil())) =  [1] x + [5]                           
                                  >= [1] x + [5]                           
                                  =  :(x,nil())                            
        
****** Step 1.b:6.a:4.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.a:4.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> sum#(xs)
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):3
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):2
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):3
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):2
          
          3:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4
          
          4:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> sum#(xs)
          2: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
          4: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
****** Step 1.b:6.a:4.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:6.a:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> +#(x,y)
            sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: +#(s(x),y) -> c_2(+#(x,y))
          
        Consider the set of all dependency pairs
          1: +#(s(x),y) -> c_2(+#(x,y))
          2: avg#(xs) -> sum#(xs)
          3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
          5: sum#(:(x,:(y,xs))) -> +#(x,y)
          6: sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.a:4.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> +#(x,y)
            sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [1] x1 + [1] x2 + [0]
               p(++) = [1] x1 + [2] x2 + [0]
                p(-) = [0]                  
                p(0) = [7]                  
                p(:) = [1] x1 + [1] x2 + [0]
              p(avg) = [0]                  
               p(hd) = [0]                  
           p(length) = [0]                  
              p(nil) = [0]                  
             p(quot) = [0]                  
                p(s) = [1] x1 + [4]         
              p(sum) = [1] x1 + [0]         
               p(+#) = [2] x1 + [1] x2 + [4]
              p(++#) = [1] x2 + [0]         
               p(-#) = [1] x1 + [0]         
             p(avg#) = [4] x1 + [4]         
              p(hd#) = [0]                  
          p(length#) = [0]                  
            p(quot#) = [0]                  
             p(sum#) = [4] x1 + [4]         
              p(c_1) = [0]                  
              p(c_2) = [1] x1 + [6]         
              p(c_3) = [0]                  
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [0]                  
              p(c_7) = [0]                  
              p(c_8) = [0]                  
              p(c_9) = [0]                  
             p(c_10) = [0]                  
             p(c_11) = [0]                  
             p(c_12) = [0]                  
             p(c_13) = [0]                  
             p(c_14) = [0]                  
             p(c_15) = [0]                  
             p(c_16) = [0]                  
        
        Following rules are strictly oriented:
        +#(s(x),y) = [2] x + [1] y + [12]
                   > [2] x + [1] y + [10]
                   = c_2(+#(x,y))        
        
        
        Following rules are (at-least) weakly oriented:
                         avg#(xs) =  [4] xs + [4]                         
                                  >= [4] xs + [4]                         
                                  =  sum#(xs)                             
        
        sum#(++(xs,:(x,:(y,ys)))) =  [8] x + [4] xs + [8] y + [8] ys + [4]
                                  >= [8] x + [4] xs + [8] y + [8] ys + [4]
                                  =  sum#(++(xs,sum(:(x,:(y,ys)))))       
        
        sum#(++(xs,:(x,:(y,ys)))) =  [8] x + [4] xs + [8] y + [8] ys + [4]
                                  >= [4] x + [4] y + [4] ys + [4]         
                                  =  sum#(:(x,:(y,ys)))                   
        
               sum#(:(x,:(y,xs))) =  [4] x + [4] xs + [4] y + [4]         
                                  >= [2] x + [1] y + [4]                  
                                  =  +#(x,y)                              
        
               sum#(:(x,:(y,xs))) =  [4] x + [4] xs + [4] y + [4]         
                                  >= [4] x + [4] xs + [4] y + [4]         
                                  =  sum#(:(+(x,y),xs))                   
        
                         +(0(),y) =  [1] y + [7]                          
                                  >= [1] y + [0]                          
                                  =  y                                    
        
                        +(s(x),y) =  [1] x + [1] y + [4]                  
                                  >= [1] x + [1] y + [4]                  
                                  =  s(+(x,y))                            
        
                   ++(:(x,xs),ys) =  [1] x + [1] xs + [2] ys + [0]        
                                  >= [1] x + [1] xs + [2] ys + [0]        
                                  =  :(x,++(xs,ys))                       
        
                     ++(nil(),ys) =  [2] ys + [0]                         
                                  >= [1] ys + [0]                         
                                  =  ys                                   
        
                sum(:(x,:(y,xs))) =  [1] x + [1] xs + [1] y + [0]         
                                  >= [1] x + [1] xs + [1] y + [0]         
                                  =  sum(:(+(x,y),xs))                    
        
                  sum(:(x,nil())) =  [1] x + [0]                          
                                  >= [1] x + [0]                          
                                  =  :(x,nil())                           
        
****** Step 1.b:6.a:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(s(x),y) -> c_2(+#(x,y))
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> +#(x,y)
            sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.a:4.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(s(x),y) -> c_2(+#(x,y))
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
            sum#(:(x,:(y,xs))) -> +#(x,y)
            sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          2:W:avg#(xs) -> sum#(xs)
             -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6
             -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):3
          
          3:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
             -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6
             -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):3
          
          4:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
             -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6
             -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5
          
          5:W:sum#(:(x,:(y,xs))) -> +#(x,y)
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):1
          
          6:W:sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
             -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6
             -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: avg#(xs) -> sum#(xs)
          3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys)))
          6: sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs))
          5: sum#(:(x,:(y,xs))) -> +#(x,y)
          1: +#(s(x),y) -> c_2(+#(x,y))
****** Step 1.b:6.a:4.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak DPs:
            +#(s(x),y) -> c_2(+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          2:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2
          
          3:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          4:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          5:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2
          
          6:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          7:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_2 +#(s(x),y) -> c_2(+#(x,y)):8
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
          
          8:W:+#(s(x),y) -> c_2(+#(x,y))
             -->_1 +#(s(x),y) -> c_2(+#(x,y)):8
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          8: +#(s(x),y) -> c_2(+#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/2,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          2:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2
          
          3:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          4:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          5:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2
          
          6:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          7:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
*** Step 1.b:6.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          - Weak DPs:
              -#(s(x),s(y)) -> c_7(-#(x,y))
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/3,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              -#(s(x),s(y)) -> c_7(-#(x,y))
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak DPs:
              ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/3,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
**** Step 1.b:6.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          2:W:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          4:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4
          
          5:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5
          
          6:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
          
          7:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: length#(:(x,xs)) -> c_10(length#(xs))
          5: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          7: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          2: -#(s(x),s(y)) -> c_7(-#(x,y))
**** Step 1.b:6.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
          
          6:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):6
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
**** Step 1.b:6.b:3.a:3: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          ++(:(x,xs),ys) -> :(x,++(xs,ys))
          ++(nil(),ys) -> ys
          sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
          sum(:(x,nil())) -> :(x,nil())
          ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
**** Step 1.b:6.b:3.a:4: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        and a lower component
          ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        Further, following extension rules are added to the lower component.
          avg#(xs) -> sum#(xs)
          sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
          sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
***** Step 1.b:6.b:3.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
          
        Consider the set of all dependency pairs
          1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
          2: avg#(xs) -> c_8(sum#(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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.b:3.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_14) = {1,2}
        
        Following symbols are considered usable:
          {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [1] x1 + [6]         
               p(++) = [4] x1 + [2] x2 + [2]
                p(-) = [4] x2 + [2]         
                p(0) = [0]                  
                p(:) = [1] x2 + [4]         
              p(avg) = [1]                  
               p(hd) = [8]                  
           p(length) = [2] x1 + [2]         
              p(nil) = [3]                  
             p(quot) = [1] x2 + [0]         
                p(s) = [1] x1 + [2]         
              p(sum) = [7]                  
               p(+#) = [2] x1 + [1] x2 + [4]
              p(++#) = [1]                  
               p(-#) = [1] x2 + [2]         
             p(avg#) = [4] x1 + [6]         
              p(hd#) = [1] x1 + [1]         
          p(length#) = [2] x1 + [8]         
            p(quot#) = [1] x2 + [0]         
             p(sum#) = [1] x1 + [0]         
              p(c_1) = [0]                  
              p(c_2) = [2] x1 + [1]         
              p(c_3) = [8] x1 + [0]         
              p(c_4) = [1]                  
              p(c_5) = [1]                  
              p(c_6) = [4]                  
              p(c_7) = [2] x1 + [1]         
              p(c_8) = [1] x1 + [6]         
              p(c_9) = [1]                  
             p(c_10) = [1] x1 + [0]         
             p(c_11) = [2]                  
             p(c_12) = [0]                  
             p(c_13) = [1] x1 + [1] x2 + [0]
             p(c_14) = [1] x1 + [1] x2 + [0]
             p(c_15) = [1]                  
             p(c_16) = [1]                  
        
        Following rules are strictly oriented:
        sum#(++(xs,:(x,:(y,ys)))) = [4] xs + [2] ys + [18]                                        
                                  > [4] xs + [17]                                                 
                                  = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        
        
        Following rules are (at-least) weakly oriented:
                 avg#(xs) =  [4] xs + [6]          
                          >= [1] xs + [6]          
                          =  c_8(sum#(xs))         
        
           ++(:(x,xs),ys) =  [4] xs + [2] ys + [18]
                          >= [4] xs + [2] ys + [6] 
                          =  :(x,++(xs,ys))        
        
             ++(nil(),ys) =  [2] ys + [14]         
                          >= [1] ys + [0]          
                          =  ys                    
        
        sum(:(x,:(y,xs))) =  [7]                   
                          >= [7]                   
                          =  sum(:(+(x,y),xs))     
        
          sum(:(x,nil())) =  [7]                   
                          >= [7]                   
                          =  :(x,nil())            
        
****** Step 1.b:6.b:3.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.b:3.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(sum#(xs))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))):2
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(sum#(xs))
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
****** Step 1.b:6.b:3.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:6.b:3.a:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          
        Consider the set of all dependency pairs
          1: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
          2: avg#(xs) -> sum#(xs)
          3: sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
          4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
        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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.b:3.a:4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak DPs:
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_3) = {1}
        
        Following symbols are considered usable:
          {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [1] x1 + [1] x2 + [2]
               p(++) = [1] x1 + [1] x2 + [0]
                p(-) = [1] x1 + [1] x2 + [0]
                p(0) = [8]                  
                p(:) = [1] x1 + [1] x2 + [2]
              p(avg) = [0]                  
               p(hd) = [1] x1 + [0]         
           p(length) = [1] x1 + [0]         
              p(nil) = [0]                  
             p(quot) = [2] x1 + [4]         
                p(s) = [13]                 
              p(sum) = [1] x1 + [0]         
               p(+#) = [2] x1 + [0]         
              p(++#) = [4] x1 + [0]         
               p(-#) = [1] x1 + [2] x2 + [4]
             p(avg#) = [5] x1 + [0]         
              p(hd#) = [1] x1 + [2]         
          p(length#) = [0]                  
            p(quot#) = [1] x1 + [1]         
             p(sum#) = [4] x1 + [0]         
              p(c_1) = [2]                  
              p(c_2) = [1]                  
              p(c_3) = [1] x1 + [4]         
              p(c_4) = [4]                  
              p(c_5) = [2]                  
              p(c_6) = [1]                  
              p(c_7) = [0]                  
              p(c_8) = [0]                  
              p(c_9) = [2]                  
             p(c_10) = [2] x1 + [0]         
             p(c_11) = [0]                  
             p(c_12) = [2]                  
             p(c_13) = [1] x1 + [2]         
             p(c_14) = [2] x1 + [1] x2 + [0]
             p(c_15) = [8]                  
             p(c_16) = [1]                  
        
        Following rules are strictly oriented:
        ++#(:(x,xs),ys) = [4] x + [4] xs + [8]
                        > [4] xs + [4]        
                        = c_3(++#(xs,ys))     
        
        
        Following rules are (at-least) weakly oriented:
                         avg#(xs) =  [5] xs + [0]                          
                                  >= [4] xs + [0]                          
                                  =  sum#(xs)                              
        
        sum#(++(xs,:(x,:(y,ys)))) =  [4] x + [4] xs + [4] y + [4] ys + [16]
                                  >= [4] xs + [0]                          
                                  =  ++#(xs,sum(:(x,:(y,ys))))             
        
        sum#(++(xs,:(x,:(y,ys)))) =  [4] x + [4] xs + [4] y + [4] ys + [16]
                                  >= [4] x + [4] xs + [4] y + [4] ys + [16]
                                  =  sum#(++(xs,sum(:(x,:(y,ys)))))        
        
                         +(0(),y) =  [1] y + [10]                          
                                  >= [1] y + [0]                           
                                  =  y                                     
        
                        +(s(x),y) =  [1] y + [15]                          
                                  >= [13]                                  
                                  =  s(+(x,y))                             
        
                   ++(:(x,xs),ys) =  [1] x + [1] xs + [1] ys + [2]         
                                  >= [1] x + [1] xs + [1] ys + [2]         
                                  =  :(x,++(xs,ys))                        
        
                     ++(nil(),ys) =  [1] ys + [0]                          
                                  >= [1] ys + [0]                          
                                  =  ys                                    
        
                sum(:(x,:(y,xs))) =  [1] x + [1] xs + [1] y + [4]          
                                  >= [1] x + [1] xs + [1] y + [4]          
                                  =  sum(:(+(x,y),xs))                     
        
                  sum(:(x,nil())) =  [1] x + [2]                           
                                  >= [1] x + [2]                           
                                  =  :(x,nil())                            
        
****** Step 1.b:6.b:3.a:4.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.b:3.a:4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
            avg#(xs) -> sum#(xs)
            sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
            sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          2:W:avg#(xs) -> sum#(xs)
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))):3
          
          3:W:sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
          
          4:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):4
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: avg#(xs) -> sum#(xs)
          4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys)))))
          3: sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys))))
          1: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
****** Step 1.b:6.b:3.a:4.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          2:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):5
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          3:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          4:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          5:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):7
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):5
          
          6:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
          
          7:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys))
             -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):7
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: ++#(:(x,xs),ys) -> c_3(++#(xs,ys))
**** Step 1.b:6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                             ,++#(xs,sum(:(x,:(y,ys))))
                                             ,sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          2:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):5
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          3:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          4:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          5:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                               ,++#(xs,sum(:(x,:(y,ys))))
                                               ,sum#(:(x,:(y,ys))))
             -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))
                                                    ,++#(xs,sum(:(x,:(y,ys))))
                                                    ,sum#(:(x,:(y,ys)))):5
          
          6:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
**** Step 1.b:6.b:3.b:3: Decompose WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              -#(s(x),s(y)) -> c_7(-#(x,y))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/2,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak DPs:
              -#(s(x),s(y)) -> c_7(-#(x,y))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/2,c_14/2,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
***** Step 1.b:6.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):5
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          3:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3
          
          4:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
          
          5:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):5
          
          6:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: length#(:(x,xs)) -> c_10(length#(xs))
          5: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          6: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
***** Step 1.b:6.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
          
          4:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
***** Step 1.b:6.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: -#(s(x),s(y)) -> c_7(-#(x,y))
          
        Consider the set of all dependency pairs
          1: -#(s(x),s(y)) -> c_7(-#(x,y))
          2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
          3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
****** Step 1.b:6.b:3.b:3.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_7) = {1},
          uargs(c_8) = {1},
          uargs(c_13) = {1,2}
        
        Following symbols are considered usable:
          {+,++,-,hd,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = x1 + x2
               p(++) = x1 + x2
                p(-) = x1     
                p(0) = 0      
                p(:) = x1 + x2
              p(avg) = 0      
               p(hd) = x1     
           p(length) = 0      
              p(nil) = 0      
             p(quot) = 0      
                p(s) = 1 + x1 
              p(sum) = x1     
               p(+#) = 0      
              p(++#) = 0      
               p(-#) = x1     
             p(avg#) = x1^2   
              p(hd#) = 0      
          p(length#) = 0      
            p(quot#) = x1^2   
             p(sum#) = 0      
              p(c_1) = 0      
              p(c_2) = 0      
              p(c_3) = 0      
              p(c_4) = 0      
              p(c_5) = 0      
              p(c_6) = 0      
              p(c_7) = x1     
              p(c_8) = x1     
              p(c_9) = 0      
             p(c_10) = 0      
             p(c_11) = 0      
             p(c_12) = 0      
             p(c_13) = x1 + x2
             p(c_14) = 0      
             p(c_15) = 0      
             p(c_16) = 0      
        
        Following rules are strictly oriented:
        -#(s(x),s(y)) = 1 + x       
                      > x           
                      = c_7(-#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
                        avg#(xs) =  xs^2                              
                                 >= xs^2                              
                                 =  c_8(quot#(hd(sum(xs)),length(xs)))
        
                quot#(s(x),s(y)) =  1 + 2*x + x^2                     
                                 >= x + x^2                           
                                 =  c_13(quot#(-(x,y),s(y)),-#(x,y))  
        
                        +(0(),y) =  y                                 
                                 >= y                                 
                                 =  y                                 
        
                       +(s(x),y) =  1 + x + y                         
                                 >= 1 + x + y                         
                                 =  s(+(x,y))                         
        
                  ++(:(x,xs),ys) =  x + xs + ys                       
                                 >= x + xs + ys                       
                                 =  :(x,++(xs,ys))                    
        
                    ++(nil(),ys) =  ys                                
                                 >= ys                                
                                 =  ys                                
        
                        -(x,0()) =  x                                 
                                 >= x                                 
                                 =  x                                 
        
                     -(0(),s(y)) =  0                                 
                                 >= 0                                 
                                 =  0()                               
        
                    -(s(x),s(y)) =  1 + x                             
                                 >= x                                 
                                 =  -(x,y)                            
        
                     hd(:(x,xs)) =  x + xs                            
                                 >= x                                 
                                 =  x                                 
        
        sum(++(xs,:(x,:(y,ys)))) =  x + xs + y + ys                   
                                 >= x + xs + y + ys                   
                                 =  sum(++(xs,sum(:(x,:(y,ys)))))     
        
               sum(:(x,:(y,xs))) =  x + xs + y                        
                                 >= x + xs + y                        
                                 =  sum(:(+(x,y),xs))                 
        
                 sum(:(x,nil())) =  x                                 
                                 >= x                                 
                                 =  :(x,nil())                        
        
****** Step 1.b:6.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

****** Step 1.b:6.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
          2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
          
          3:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
          3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
          1: -#(s(x),s(y)) -> c_7(-#(x,y))
****** Step 1.b:6.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

***** Step 1.b:6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            -#(s(x),s(y)) -> c_7(-#(x,y))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          2:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          3:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):6
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
          
          4:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
          
          5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
          
          6:W:-#(s(x),s(y)) -> c_7(-#(x,y))
             -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):6
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: -#(s(x),s(y)) -> c_7(-#(x,y))
***** Step 1.b:6.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          2:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          3:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3
          
          4:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
          
          5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
***** Step 1.b:6.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak DPs:
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/2,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              length#(:(x,xs)) -> c_10(length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/2,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
****** Step 1.b:6.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          2:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          3:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3
          
          4:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
          
          5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: length#(:(x,xs)) -> c_10(length#(xs))
          3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
****** Step 1.b:6.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
          
          4:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
          
          5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(sum#(xs))
****** Step 1.b:6.b:3.b:3.b:3.a:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          ++(:(x,xs),ys) -> :(x,++(xs,ys))
          ++(nil(),ys) -> ys
          sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
          sum(:(x,nil())) -> :(x,nil())
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
****** Step 1.b:6.b:3.b:3.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: avg#(xs) -> c_8(sum#(xs))
          2: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
          
        The strictly oriented rules are moved into the weak component.
******* Step 1.b:6.b:3.b:3.b:3.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_14) = {1,2},
          uargs(c_15) = {1}
        
        Following symbols are considered usable:
          {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [8]                  
               p(++) = [1] x1 + [2] x2 + [0]
                p(-) = [1] x1 + [8] x2 + [1]
                p(0) = [4]                  
                p(:) = [1] x2 + [1]         
              p(avg) = [1] x1 + [0]         
               p(hd) = [1] x1 + [1]         
           p(length) = [2] x1 + [2]         
              p(nil) = [0]                  
             p(quot) = [2] x2 + [1]         
                p(s) = [1] x1 + [14]        
              p(sum) = [1]                  
               p(+#) = [1] x1 + [1] x2 + [1]
              p(++#) = [1] x1 + [1] x2 + [0]
               p(-#) = [1]                  
             p(avg#) = [8] x1 + [9]         
              p(hd#) = [1] x1 + [4]         
          p(length#) = [2] x1 + [1]         
            p(quot#) = [1] x1 + [2]         
             p(sum#) = [2] x1 + [0]         
              p(c_1) = [2]                  
              p(c_2) = [0]                  
              p(c_3) = [1] x1 + [1]         
              p(c_4) = [1]                  
              p(c_5) = [2]                  
              p(c_6) = [2]                  
              p(c_7) = [1] x1 + [1]         
              p(c_8) = [4] x1 + [2]         
              p(c_9) = [0]                  
             p(c_10) = [1] x1 + [8]         
             p(c_11) = [0]                  
             p(c_12) = [0]                  
             p(c_13) = [2] x1 + [2]         
             p(c_14) = [1] x1 + [1] x2 + [0]
             p(c_15) = [1] x1 + [0]         
             p(c_16) = [0]                  
        
        Following rules are strictly oriented:
                  avg#(xs) = [8] xs + [9]            
                           > [8] xs + [2]            
                           = c_8(sum#(xs))           
        
        sum#(:(x,:(y,xs))) = [2] xs + [4]            
                           > [2] xs + [2]            
                           = c_15(sum#(:(+(x,y),xs)))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(++(xs,:(x,:(y,ys)))) =  [2] xs + [4] ys + [8]                                  
                                  >= [2] xs + [2] ys + [8]                                  
                                  =  c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        
                   ++(:(x,xs),ys) =  [1] xs + [2] ys + [1]                                  
                                  >= [1] xs + [2] ys + [1]                                  
                                  =  :(x,++(xs,ys))                                         
        
                     ++(nil(),ys) =  [2] ys + [0]                                           
                                  >= [1] ys + [0]                                           
                                  =  ys                                                     
        
                sum(:(x,:(y,xs))) =  [1]                                                    
                                  >= [1]                                                    
                                  =  sum(:(+(x,y),xs))                                      
        
                  sum(:(x,nil())) =  [1]                                                    
                                  >= [1]                                                    
                                  =  :(x,nil())                                             
        
******* Step 1.b:6.b:3.b:3.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******* Step 1.b:6.b:3.b:3.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(sum#(xs))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2
          
          3:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(sum#(xs))
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
          3: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
******* Step 1.b:6.b:3.b:3.b:3.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

****** Step 1.b:6.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1
          
          2:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
          
          3:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1
          
          5:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
             -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
****** Step 1.b:6.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1
          
          2:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
          
          3:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
****** Step 1.b:6.b:3.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              length#(:(x,xs)) -> c_10(length#(xs))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/1,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
              length#(:(x,xs)) -> c_10(length#(xs))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/1,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
******* Step 1.b:6.b:3.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1
          
          2:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
          
          3:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):3
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          2: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
******* Step 1.b:6.b:3.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1
          
          4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(length#(xs))
******* Step 1.b:6.b:3.b:3.b:3.b:3.a:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak DPs:
            avg#(xs) -> c_8(length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          avg#(xs) -> c_8(length#(xs))
          length#(:(x,xs)) -> c_10(length#(xs))
******* Step 1.b:6.b:3.b:3.b:3.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak DPs:
            avg#(xs) -> c_8(length#(xs))
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: length#(:(x,xs)) -> c_10(length#(xs))
          
        Consider the set of all dependency pairs
          1: length#(:(x,xs)) -> c_10(length#(xs))
          2: avg#(xs) -> c_8(length#(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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
******** Step 1.b:6.b:3.b:3.b:3.b:3.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak DPs:
            avg#(xs) -> c_8(length#(xs))
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [0]                  
               p(++) = [0]                  
                p(-) = [0]                  
                p(0) = [0]                  
                p(:) = [1] x1 + [1] x2 + [4]
              p(avg) = [0]                  
               p(hd) = [0]                  
           p(length) = [0]                  
              p(nil) = [0]                  
             p(quot) = [0]                  
                p(s) = [1] x1 + [0]         
              p(sum) = [0]                  
               p(+#) = [0]                  
              p(++#) = [0]                  
               p(-#) = [0]                  
             p(avg#) = [8] x1 + [0]         
              p(hd#) = [0]                  
          p(length#) = [1] x1 + [0]         
            p(quot#) = [0]                  
             p(sum#) = [0]                  
              p(c_1) = [0]                  
              p(c_2) = [0]                  
              p(c_3) = [0]                  
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [0]                  
              p(c_7) = [0]                  
              p(c_8) = [8] x1 + [0]         
              p(c_9) = [0]                  
             p(c_10) = [1] x1 + [0]         
             p(c_11) = [0]                  
             p(c_12) = [0]                  
             p(c_13) = [0]                  
             p(c_14) = [0]                  
             p(c_15) = [0]                  
             p(c_16) = [0]                  
        
        Following rules are strictly oriented:
        length#(:(x,xs)) = [1] x + [1] xs + [4]
                         > [1] xs + [0]        
                         = c_10(length#(xs))   
        
        
        Following rules are (at-least) weakly oriented:
        avg#(xs) =  [8] xs + [0]    
                 >= [8] xs + [0]    
                 =  c_8(length#(xs))
        
******** Step 1.b:6.b:3.b:3.b:3.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

******** Step 1.b:6.b:3.b:3.b:3.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2
          
          2:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(length#(xs))
          2: length#(:(x,xs)) -> c_10(length#(xs))
******** Step 1.b:6.b:3.b:3.b:3.b:3.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

******* Step 1.b:6.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
            length#(:(x,xs)) -> c_10(length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
          2:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
          4:W:length#(:(x,xs)) -> c_10(length#(xs))
             -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: length#(:(x,xs)) -> c_10(length#(xs))
******* Step 1.b:6.b:3.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
          2:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
******* Step 1.b:6.b:3.b:3.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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:
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/1,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
        
        Problem (S)
          - Strict DPs:
              sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          - Weak DPs:
              avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
              quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
          - Weak TRS:
              +(0(),y) -> y
              +(s(x),y) -> s(+(x,y))
              ++(:(x,xs),ys) -> :(x,++(xs,ys))
              ++(nil(),ys) -> ys
              -(x,0()) -> x
              -(0(),s(y)) -> 0()
              -(s(x),s(y)) -> -(x,y)
              hd(:(x,xs)) -> x
              length(:(x,xs)) -> s(length(xs))
              length(nil()) -> 0()
              sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
              sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
              sum(:(x,nil())) -> :(x,nil())
          - Signature:
              {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2
              ,sum#/1} / {0/0,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0
              ,c_13/1,c_14/1,c_15/1,c_16/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
              ,sum#} and constructors {0,:,nil,s}
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
          3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
          
        Consider the set of all dependency pairs
          1: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
          2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_13) = {1}
        
        Following symbols are considered usable:
          {+,++,-,hd,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [1] x1 + [1] x2 + [0]
               p(++) = [2] x1 + [1] x2 + [8]
                p(-) = [1] x1 + [0]         
                p(0) = [0]                  
                p(:) = [1] x1 + [1] x2 + [0]
              p(avg) = [2] x1 + [2]         
               p(hd) = [1] x1 + [0]         
           p(length) = [8]                  
              p(nil) = [0]                  
             p(quot) = [4]                  
                p(s) = [1] x1 + [2]         
              p(sum) = [1] x1 + [0]         
               p(+#) = [2] x2 + [0]         
              p(++#) = [2] x1 + [1] x2 + [0]
               p(-#) = [1]                  
             p(avg#) = [12] x1 + [3]        
              p(hd#) = [1]                  
          p(length#) = [2]                  
            p(quot#) = [12] x1 + [2]        
             p(sum#) = [1]                  
              p(c_1) = [0]                  
              p(c_2) = [1]                  
              p(c_3) = [2]                  
              p(c_4) = [1]                  
              p(c_5) = [1]                  
              p(c_6) = [1]                  
              p(c_7) = [0]                  
              p(c_8) = [1] x1 + [1]         
              p(c_9) = [1]                  
             p(c_10) = [1] x1 + [1]         
             p(c_11) = [1]                  
             p(c_12) = [1]                  
             p(c_13) = [1] x1 + [0]         
             p(c_14) = [1] x1 + [4]         
             p(c_15) = [0]                  
             p(c_16) = [1]                  
        
        Following rules are strictly oriented:
        quot#(s(x),s(y)) = [12] x + [26]           
                         > [12] x + [2]            
                         = c_13(quot#(-(x,y),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
                        avg#(xs) =  [12] xs + [3]                        
                                 >= [12] xs + [3]                        
                                 =  c_8(quot#(hd(sum(xs)),length(xs)))   
        
                        +(0(),y) =  [1] y + [0]                          
                                 >= [1] y + [0]                          
                                 =  y                                    
        
                       +(s(x),y) =  [1] x + [1] y + [2]                  
                                 >= [1] x + [1] y + [2]                  
                                 =  s(+(x,y))                            
        
                  ++(:(x,xs),ys) =  [2] x + [2] xs + [1] ys + [8]        
                                 >= [1] x + [2] xs + [1] ys + [8]        
                                 =  :(x,++(xs,ys))                       
        
                    ++(nil(),ys) =  [1] ys + [8]                         
                                 >= [1] ys + [0]                         
                                 =  ys                                   
        
                        -(x,0()) =  [1] x + [0]                          
                                 >= [1] x + [0]                          
                                 =  x                                    
        
                     -(0(),s(y)) =  [0]                                  
                                 >= [0]                                  
                                 =  0()                                  
        
                    -(s(x),s(y)) =  [1] x + [2]                          
                                 >= [1] x + [0]                          
                                 =  -(x,y)                               
        
                     hd(:(x,xs)) =  [1] x + [1] xs + [0]                 
                                 >= [1] x + [0]                          
                                 =  x                                    
        
        sum(++(xs,:(x,:(y,ys)))) =  [1] x + [2] xs + [1] y + [1] ys + [8]
                                 >= [1] x + [2] xs + [1] y + [1] ys + [8]
                                 =  sum(++(xs,sum(:(x,:(y,ys)))))        
        
               sum(:(x,:(y,xs))) =  [1] x + [1] xs + [1] y + [0]         
                                 >= [1] x + [1] xs + [1] y + [0]         
                                 =  sum(:(+(x,y),xs))                    
        
                 sum(:(x,nil())) =  [1] x + [0]                          
                                 >= [1] x + [0]                          
                                 =  :(x,nil())                           
        
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
          
          2:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
          2: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
            quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1
          
          2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1
          
          3:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
             -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1
          
          2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
             -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          avg#(xs) -> c_8(sum#(xs))
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            -(x,0()) -> x
            -(0(),s(y)) -> 0()
            -(s(x),s(y)) -> -(x,y)
            hd(:(x,xs)) -> x
            length(:(x,xs)) -> s(length(xs))
            length(nil()) -> 0()
            sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys)))))
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          ++(:(x,xs),ys) -> :(x,++(xs,ys))
          ++(nil(),ys) -> ys
          sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
          sum(:(x,nil())) -> :(x,nil())
          avg#(xs) -> c_8(sum#(xs))
          sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
******** Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          
        Consider the set of all dependency pairs
          1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
          2: avg#(xs) -> c_8(sum#(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}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + 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_8) = {1},
          uargs(c_14) = {1}
        
        Following symbols are considered usable:
          {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
        TcT has computed the following interpretation:
                p(+) = [7] x1 + [1] x2 + [1]
               p(++) = [4] x1 + [4] x2 + [0]
                p(-) = [4] x1 + [2] x2 + [1]
                p(0) = [2]                  
                p(:) = [1] x2 + [1]         
              p(avg) = [2] x1 + [0]         
               p(hd) = [0]                  
           p(length) = [2] x1 + [2]         
              p(nil) = [0]                  
             p(quot) = [1] x1 + [2] x2 + [1]
                p(s) = [1] x1 + [2]         
              p(sum) = [1]                  
               p(+#) = [1] x1 + [2] x2 + [1]
              p(++#) = [1] x1 + [1] x2 + [2]
               p(-#) = [1] x1 + [1] x2 + [0]
             p(avg#) = [8] x1 + [12]        
              p(hd#) = [0]                  
          p(length#) = [4] x1 + [1]         
            p(quot#) = [8] x1 + [1] x2 + [1]
             p(sum#) = [2] x1 + [8]         
              p(c_1) = [2]                  
              p(c_2) = [8] x1 + [0]         
              p(c_3) = [1] x1 + [1]         
              p(c_4) = [2]                  
              p(c_5) = [0]                  
              p(c_6) = [1]                  
              p(c_7) = [8]                  
              p(c_8) = [1] x1 + [4]         
              p(c_9) = [4]                  
             p(c_10) = [2]                  
             p(c_11) = [1]                  
             p(c_12) = [0]                  
             p(c_13) = [1] x1 + [1]         
             p(c_14) = [1] x1 + [0]         
             p(c_15) = [2] x1 + [1]         
             p(c_16) = [2]                  
        
        Following rules are strictly oriented:
        sum#(++(xs,:(x,:(y,ys)))) = [8] xs + [8] ys + [24]              
                                  > [8] xs + [16]                       
                                  = c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        
        
        Following rules are (at-least) weakly oriented:
                 avg#(xs) =  [8] xs + [12]        
                          >= [2] xs + [12]        
                          =  c_8(sum#(xs))        
        
           ++(:(x,xs),ys) =  [4] xs + [4] ys + [4]
                          >= [4] xs + [4] ys + [1]
                          =  :(x,++(xs,ys))       
        
             ++(nil(),ys) =  [4] ys + [0]         
                          >= [1] ys + [0]         
                          =  ys                   
        
        sum(:(x,:(y,xs))) =  [1]                  
                          >= [1]                  
                          =  sum(:(+(x,y),xs))    
        
          sum(:(x,nil())) =  [1]                  
                          >= [1]                  
                          =  :(x,nil())           
        
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            avg#(xs) -> c_8(sum#(xs))
            sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:avg#(xs) -> c_8(sum#(xs))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
          2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
             -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: avg#(xs) -> c_8(sum#(xs))
          2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
********* Step 1.b:6.b:3.b:3.b:3.b:3.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            ++(:(x,xs),ys) -> :(x,++(xs,ys))
            ++(nil(),ys) -> ys
            sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs))
            sum(:(x,nil())) -> :(x,nil())
        - Signature:
            {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0
            ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1
            ,c_15/1,c_16/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot#
            ,sum#} and constructors {0,:,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))