MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            append(cons(x,xs),ys()) -> cons(x,append(xs,ys()))
            append(nil(),ys()) -> ys()
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append,filterhigh,filterlow,ge,if1,if2
            ,qsort} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
          append#(nil(),ys()) -> c_2()
          filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
          filterhigh#(n,nil()) -> c_4()
          filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
          filterlow#(n,nil()) -> c_6()
          ge#(x,0()) -> c_7()
          ge#(0(),s(x)) -> c_8()
          ge#(s(x),s(y)) -> c_9(ge#(x,y))
          if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
          if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
          if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
          if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
          qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                    ,qsort#(filterlow(x,cons(x,xs)))
                                    ,filterlow#(x,cons(x,xs))
                                    ,qsort#(filterhigh(x,cons(x,xs)))
                                    ,filterhigh#(x,cons(x,xs)))
          qsort#(nil()) -> c_15()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
            append#(nil(),ys()) -> c_2()
            filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
            filterhigh#(n,nil()) -> c_4()
            filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
            filterlow#(n,nil()) -> c_6()
            ge#(x,0()) -> c_7()
            ge#(0(),s(x)) -> c_8()
            ge#(s(x),s(y)) -> c_9(ge#(x,y))
            if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
            if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
            if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
            if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
            qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                      ,qsort#(filterlow(x,cons(x,xs)))
                                      ,filterlow#(x,cons(x,xs))
                                      ,qsort#(filterhigh(x,cons(x,xs)))
                                      ,filterhigh#(x,cons(x,xs)))
            qsort#(nil()) -> c_15()
        - Weak TRS:
            append(cons(x,xs),ys()) -> cons(x,append(xs,ys()))
            append(nil(),ys()) -> ys()
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
            ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
            ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/5,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
            ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
          filterhigh(n,nil()) -> nil()
          filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
          filterlow(n,nil()) -> nil()
          ge(x,0()) -> true()
          ge(0(),s(x)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
          if1(true(),n,x,xs) -> filterlow(n,xs)
          if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
          if2(true(),n,x,xs) -> filterhigh(n,xs)
          qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
          qsort(nil()) -> nil()
          append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
          append#(nil(),ys()) -> c_2()
          filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
          filterhigh#(n,nil()) -> c_4()
          filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
          filterlow#(n,nil()) -> c_6()
          ge#(x,0()) -> c_7()
          ge#(0(),s(x)) -> c_8()
          ge#(s(x),s(y)) -> c_9(ge#(x,y))
          if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
          if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
          if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
          if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
          qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                    ,qsort#(filterlow(x,cons(x,xs)))
                                    ,filterlow#(x,cons(x,xs))
                                    ,qsort#(filterhigh(x,cons(x,xs)))
                                    ,filterhigh#(x,cons(x,xs)))
          qsort#(nil()) -> c_15()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
            append#(nil(),ys()) -> c_2()
            filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
            filterhigh#(n,nil()) -> c_4()
            filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
            filterlow#(n,nil()) -> c_6()
            ge#(x,0()) -> c_7()
            ge#(0(),s(x)) -> c_8()
            ge#(s(x),s(y)) -> c_9(ge#(x,y))
            if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
            if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
            if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
            if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
            qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                      ,qsort#(filterlow(x,cons(x,xs)))
                                      ,filterlow#(x,cons(x,xs))
                                      ,qsort#(filterhigh(x,cons(x,xs)))
                                      ,filterhigh#(x,cons(x,xs)))
            qsort#(nil()) -> c_15()
        - Weak TRS:
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
            ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
            ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/5,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
            ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,6,7,8,15}
        by application of
          Pre({2,4,6,7,8,15}) = {1,3,5,9,10,11,12,13,14}.
        Here rules are labelled as follows:
          1: append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
          2: append#(nil(),ys()) -> c_2()
          3: filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
          4: filterhigh#(n,nil()) -> c_4()
          5: filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
          6: filterlow#(n,nil()) -> c_6()
          7: ge#(x,0()) -> c_7()
          8: ge#(0(),s(x)) -> c_8()
          9: ge#(s(x),s(y)) -> c_9(ge#(x,y))
          10: if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
          11: if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
          12: if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
          13: if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
          14: qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                        ,qsort#(filterlow(x,cons(x,xs)))
                                        ,filterlow#(x,cons(x,xs))
                                        ,qsort#(filterhigh(x,cons(x,xs)))
                                        ,filterhigh#(x,cons(x,xs)))
          15: qsort#(nil()) -> c_15()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
            filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
            filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
            ge#(s(x),s(y)) -> c_9(ge#(x,y))
            if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
            if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
            if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
            if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
            qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                      ,qsort#(filterlow(x,cons(x,xs)))
                                      ,filterlow#(x,cons(x,xs))
                                      ,qsort#(filterhigh(x,cons(x,xs)))
                                      ,filterhigh#(x,cons(x,xs)))
        - Weak DPs:
            append#(nil(),ys()) -> c_2()
            filterhigh#(n,nil()) -> c_4()
            filterlow#(n,nil()) -> c_6()
            ge#(x,0()) -> c_7()
            ge#(0(),s(x)) -> c_8()
            qsort#(nil()) -> c_15()
        - Weak TRS:
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
            ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
            ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/5,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
            ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
             -->_1 append#(nil(),ys()) -> c_2():10
             -->_1 append#(cons(x,xs),ys()) -> c_1(append#(xs,ys())):1
          
          2:S:filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
             -->_1 if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs)):8
             -->_1 if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs)):7
             -->_2 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
             -->_2 ge#(0(),s(x)) -> c_8():14
             -->_2 ge#(x,0()) -> c_7():13
          
          3:S:filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
             -->_1 if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs)):6
             -->_1 if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs)):5
             -->_2 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
             -->_2 ge#(0(),s(x)) -> c_8():14
             -->_2 ge#(x,0()) -> c_7():13
          
          4:S:ge#(s(x),s(y)) -> c_9(ge#(x,y))
             -->_1 ge#(0(),s(x)) -> c_8():14
             -->_1 ge#(x,0()) -> c_7():13
             -->_1 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
          
          5:S:if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
             -->_1 filterlow#(n,nil()) -> c_6():12
             -->_1 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
          
          6:S:if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
             -->_1 filterlow#(n,nil()) -> c_6():12
             -->_1 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
          
          7:S:if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
             -->_1 filterhigh#(n,nil()) -> c_4():11
             -->_1 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
          8:S:if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
             -->_1 filterhigh#(n,nil()) -> c_4():11
             -->_1 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
          9:S:qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                        ,qsort#(filterlow(x,cons(x,xs)))
                                        ,filterlow#(x,cons(x,xs))
                                        ,qsort#(filterhigh(x,cons(x,xs)))
                                        ,filterhigh#(x,cons(x,xs)))
             -->_4 qsort#(nil()) -> c_15():15
             -->_2 qsort#(nil()) -> c_15():15
             -->_4 qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                     ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                             ,qsort#(filterlow(x,cons(x,xs)))
                                             ,filterlow#(x,cons(x,xs))
                                             ,qsort#(filterhigh(x,cons(x,xs)))
                                             ,filterhigh#(x,cons(x,xs))):9
             -->_2 qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                     ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                             ,qsort#(filterlow(x,cons(x,xs)))
                                             ,filterlow#(x,cons(x,xs))
                                             ,qsort#(filterhigh(x,cons(x,xs)))
                                             ,filterhigh#(x,cons(x,xs))):9
             -->_3 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
             -->_5 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
          10:W:append#(nil(),ys()) -> c_2()
             
          
          11:W:filterhigh#(n,nil()) -> c_4()
             
          
          12:W:filterlow#(n,nil()) -> c_6()
             
          
          13:W:ge#(x,0()) -> c_7()
             
          
          14:W:ge#(0(),s(x)) -> c_8()
             
          
          15:W:qsort#(nil()) -> c_15()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          15: qsort#(nil()) -> c_15()
          12: filterlow#(n,nil()) -> c_6()
          13: ge#(x,0()) -> c_7()
          14: ge#(0(),s(x)) -> c_8()
          11: filterhigh#(n,nil()) -> c_4()
          10: append#(nil(),ys()) -> c_2()
* Step 5: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
            filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
            filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
            ge#(s(x),s(y)) -> c_9(ge#(x,y))
            if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
            if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
            if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
            if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
            qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                      ,qsort#(filterlow(x,cons(x,xs)))
                                      ,filterlow#(x,cons(x,xs))
                                      ,qsort#(filterhigh(x,cons(x,xs)))
                                      ,filterhigh#(x,cons(x,xs)))
        - Weak TRS:
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
            ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
            ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/5,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
            ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
             -->_1 append#(cons(x,xs),ys()) -> c_1(append#(xs,ys())):1
          
          2:S:filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
             -->_1 if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs)):8
             -->_1 if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs)):7
             -->_2 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
          
          3:S:filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
             -->_1 if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs)):6
             -->_1 if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs)):5
             -->_2 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
          
          4:S:ge#(s(x),s(y)) -> c_9(ge#(x,y))
             -->_1 ge#(s(x),s(y)) -> c_9(ge#(x,y)):4
          
          5:S:if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
             -->_1 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
          
          6:S:if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
             -->_1 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
          
          7:S:if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
             -->_1 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
          8:S:if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
             -->_1 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
          9:S:qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                        ,qsort#(filterlow(x,cons(x,xs)))
                                        ,filterlow#(x,cons(x,xs))
                                        ,qsort#(filterhigh(x,cons(x,xs)))
                                        ,filterhigh#(x,cons(x,xs)))
             -->_4 qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                     ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                             ,qsort#(filterlow(x,cons(x,xs)))
                                             ,filterlow#(x,cons(x,xs))
                                             ,qsort#(filterhigh(x,cons(x,xs)))
                                             ,filterhigh#(x,cons(x,xs))):9
             -->_2 qsort#(cons(x,xs)) -> c_14(append#(qsort(filterlow(x,cons(x,xs)))
                                                     ,cons(x,qsort(filterhigh(x,cons(x,xs)))))
                                             ,qsort#(filterlow(x,cons(x,xs)))
                                             ,filterlow#(x,cons(x,xs))
                                             ,qsort#(filterhigh(x,cons(x,xs)))
                                             ,filterhigh#(x,cons(x,xs))):9
             -->_3 filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x)):3
             -->_5 filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          qsort#(cons(x,xs)) -> c_14(qsort#(filterlow(x,cons(x,xs)))
                                    ,filterlow#(x,cons(x,xs))
                                    ,qsort#(filterhigh(x,cons(x,xs)))
                                    ,filterhigh#(x,cons(x,xs)))
* Step 6: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
            filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
            filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
            ge#(s(x),s(y)) -> c_9(ge#(x,y))
            if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
            if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
            if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
            if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
            qsort#(cons(x,xs)) -> c_14(qsort#(filterlow(x,cons(x,xs)))
                                      ,filterlow#(x,cons(x,xs))
                                      ,qsort#(filterhigh(x,cons(x,xs)))
                                      ,filterhigh#(x,cons(x,xs)))
        - Weak TRS:
            filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
            filterhigh(n,nil()) -> nil()
            filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
            filterlow(n,nil()) -> nil()
            ge(x,0()) -> true()
            ge(0(),s(x)) -> false()
            ge(s(x),s(y)) -> ge(x,y)
            if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
            if1(true(),n,x,xs) -> filterlow(n,xs)
            if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
            if2(true(),n,x,xs) -> filterhigh(n,xs)
            qsort(cons(x,xs)) -> append(qsort(filterlow(x,cons(x,xs))),cons(x,qsort(filterhigh(x,cons(x,xs)))))
            qsort(nil()) -> nil()
        - Signature:
            {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
            ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
            ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/4,c_15/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
            ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
          filterhigh(n,nil()) -> nil()
          filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
          filterlow(n,nil()) -> nil()
          ge(x,0()) -> true()
          ge(0(),s(x)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
          if1(true(),n,x,xs) -> filterlow(n,xs)
          if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
          if2(true(),n,x,xs) -> filterhigh(n,xs)
          append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
          filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
          filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
          ge#(s(x),s(y)) -> c_9(ge#(x,y))
          if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
          if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
          if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
          if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
          qsort#(cons(x,xs)) -> c_14(qsort#(filterlow(x,cons(x,xs)))
                                    ,filterlow#(x,cons(x,xs))
                                    ,qsort#(filterhigh(x,cons(x,xs)))
                                    ,filterhigh#(x,cons(x,xs)))
* Step 7: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          append#(cons(x,xs),ys()) -> c_1(append#(xs,ys()))
          filterhigh#(n,cons(x,xs)) -> c_3(if2#(ge(x,n),n,x,xs),ge#(x,n))
          filterlow#(n,cons(x,xs)) -> c_5(if1#(ge(n,x),n,x,xs),ge#(n,x))
          ge#(s(x),s(y)) -> c_9(ge#(x,y))
          if1#(false(),n,x,xs) -> c_10(filterlow#(n,xs))
          if1#(true(),n,x,xs) -> c_11(filterlow#(n,xs))
          if2#(false(),n,x,xs) -> c_12(filterhigh#(n,xs))
          if2#(true(),n,x,xs) -> c_13(filterhigh#(n,xs))
          qsort#(cons(x,xs)) -> c_14(qsort#(filterlow(x,cons(x,xs)))
                                    ,filterlow#(x,cons(x,xs))
                                    ,qsort#(filterhigh(x,cons(x,xs)))
                                    ,filterhigh#(x,cons(x,xs)))
      - Weak TRS:
          filterhigh(n,cons(x,xs)) -> if2(ge(x,n),n,x,xs)
          filterhigh(n,nil()) -> nil()
          filterlow(n,cons(x,xs)) -> if1(ge(n,x),n,x,xs)
          filterlow(n,nil()) -> nil()
          ge(x,0()) -> true()
          ge(0(),s(x)) -> false()
          ge(s(x),s(y)) -> ge(x,y)
          if1(false(),n,x,xs) -> cons(x,filterlow(n,xs))
          if1(true(),n,x,xs) -> filterlow(n,xs)
          if2(false(),n,x,xs) -> cons(x,filterhigh(n,xs))
          if2(true(),n,x,xs) -> filterhigh(n,xs)
      - Signature:
          {append/2,filterhigh/2,filterlow/2,ge/2,if1/4,if2/4,qsort/1,append#/2,filterhigh#/2,filterlow#/2,ge#/2
          ,if1#/4,if2#/4,qsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,ys/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/2,c_6/0
          ,c_7/0,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1,c_14/4,c_15/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {append#,filterhigh#,filterlow#,ge#,if1#,if2#
          ,qsort#} and constructors {0,cons,false,nil,s,true,ys}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
MAYBE