MAYBE
* Step 1: DependencyPairs MAYBE
    + Considered Problem:
        - Strict TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
            qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
            qsort(nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1} / {++/2,./2,<=/2,if/3,nil/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters,lowers,qsort} and constructors {++,.,<=,if,nil}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
          greaters#(x,nil()) -> c_2()
          lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
          lowers#(x,nil()) -> c_4()
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
          qsort#(nil()) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
            greaters#(x,nil()) -> c_2()
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            lowers#(x,nil()) -> c_4()
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
            qsort#(nil()) -> c_6()
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
            qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
            qsort(nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
          greaters(x,nil()) -> nil()
          lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
          lowers(x,nil()) -> nil()
          greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
          greaters#(x,nil()) -> c_2()
          lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
          lowers#(x,nil()) -> c_4()
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
          qsort#(nil()) -> c_6()
* Step 3: PredecessorEstimation MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
            greaters#(x,nil()) -> c_2()
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            lowers#(x,nil()) -> c_4()
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
            qsort#(nil()) -> c_6()
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4,6}
        by application of
          Pre({2,4,6}) = {1,3,5}.
        Here rules are labelled as follows:
          1: greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
          2: greaters#(x,nil()) -> c_2()
          3: lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
          4: lowers#(x,nil()) -> c_4()
          5: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
          6: qsort#(nil()) -> c_6()
* Step 4: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak DPs:
            greaters#(x,nil()) -> c_2()
            lowers#(x,nil()) -> c_4()
            qsort#(nil()) -> c_6()
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
             -->_2 greaters#(x,nil()) -> c_2():4
             -->_1 greaters#(x,nil()) -> c_2():4
             -->_2 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
             -->_1 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
          
          2:S:lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
             -->_2 lowers#(x,nil()) -> c_4():5
             -->_1 lowers#(x,nil()) -> c_4():5
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_1 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
          
          3:S:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
             -->_3 qsort#(nil()) -> c_6():6
             -->_1 qsort#(nil()) -> c_6():6
             -->_2 lowers#(x,nil()) -> c_4():5
             -->_4 greaters#(x,nil()) -> c_2():4
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_4 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
          
          4:W:greaters#(x,nil()) -> c_2()
             
          
          5:W:lowers#(x,nil()) -> c_4()
             
          
          6:W:qsort#(nil()) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: qsort#(nil()) -> c_6()
          5: lowers#(x,nil()) -> c_4()
          4: greaters#(x,nil()) -> c_2()
* Step 5: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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:
              greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
          - Weak DPs:
              lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
              qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
          - Weak TRS:
              greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
              greaters(x,nil()) -> nil()
              lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
              lowers(x,nil()) -> nil()
          - Signature:
              {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
              ,c_4/0,c_5/4,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
              ,nil}
        
        Problem (S)
          - Strict DPs:
              lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
              qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
          - Weak DPs:
              greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
          - Weak TRS:
              greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
              greaters(x,nil()) -> nil()
              lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
              lowers(x,nil()) -> nil()
          - Signature:
              {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
              ,c_4/0,c_5/4,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
              ,nil}
** Step 5.a:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
        - Weak DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
             -->_2 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
             -->_1 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
          
          2:W:lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_1 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
          
          3:W:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
             -->_4 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
** Step 5.a:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
             -->_2 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
             -->_1 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
          
          3:W:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
             -->_4 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):1
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
** Step 5.a:3: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        and a lower component
          greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
        Further, following extension rules are added to the lower component.
          qsort#(.(x,y)) -> greaters#(x,y)
          qsort#(.(x,y)) -> qsort#(greaters(x,y))
          qsort#(.(x,y)) -> qsort#(lowers(x,y))
*** Step 5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1,2}
        
        Following symbols are considered usable:
          {greaters,lowers,greaters#,lowers#,qsort#}
        TcT has computed the following interpretation:
                 p(++) = [1] x2 + [0]         
                  p(.) = [1] x1 + [9]         
                 p(<=) = [1] x1 + [4]         
           p(greaters) = [4]                  
                 p(if) = [0]                  
             p(lowers) = [0]                  
                p(nil) = [0]                  
              p(qsort) = [1] x1 + [8]         
          p(greaters#) = [4] x1 + [8]         
            p(lowers#) = [1]                  
             p(qsort#) = [1] x1 + [0]         
                p(c_1) = [1] x1 + [1]         
                p(c_2) = [1]                  
                p(c_3) = [0]                  
                p(c_4) = [1]                  
                p(c_5) = [4] x1 + [2] x2 + [0]
                p(c_6) = [2]                  
        
        Following rules are strictly oriented:
        qsort#(.(x,y)) = [1] x + [9]                                                  
                       > [8]                                                          
                       = c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        greaters(x,.(y,z)) =  [4]                                         
                           >= [0]                                         
                           =  if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
        
         greaters(x,nil()) =  [4]                                         
                           >= [0]                                         
                           =  nil()                                       
        
          lowers(x,.(y,z)) =  [0]                                         
                           >= [0]                                         
                           =  if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))    
        
           lowers(x,nil()) =  [0]                                         
                           >= [0]                                         
                           =  nil()                                       
        
**** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
             -->_2 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y)):1
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)),greaters#(x,y))
**** Step 5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 5.a:3.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
      - Weak DPs:
          qsort#(.(x,y)) -> greaters#(x,y)
          qsort#(.(x,y)) -> qsort#(greaters(x,y))
          qsort#(.(x,y)) -> qsort#(lowers(x,y))
      - Weak TRS:
          greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
          greaters(x,nil()) -> nil()
          lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
          lowers(x,nil()) -> nil()
      - Signature:
          {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
          ,c_4/0,c_5/3,c_6/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
          ,nil}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
** Step 5.b:1: RemoveWeakSuffixes MAYBE
    + Considered Problem:
        - Strict DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak DPs:
            greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
             -->_1 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
          
          2:S:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
             -->_4 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):3
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):2
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):2
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
          
          3:W:greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
             -->_2 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):3
             -->_1 greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: greaters#(x,.(y,z)) -> c_1(greaters#(x,z),greaters#(x,z))
** Step 5.b:2: SimplifyRHS MAYBE
    + Considered Problem:
        - Strict DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/4,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
             -->_1 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
          
          2:S:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y))
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):2
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)),greaters#(x,y)):2
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
** Step 5.b:3: Decompose MAYBE
    + Considered Problem:
        - Strict DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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:
              lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
          - Weak DPs:
              qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
          - Weak TRS:
              greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
              greaters(x,nil()) -> nil()
              lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
              lowers(x,nil()) -> nil()
          - Signature:
              {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
              ,c_4/0,c_5/3,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
              ,nil}
        
        Problem (S)
          - Strict DPs:
              qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
          - Weak DPs:
              lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
          - Weak TRS:
              greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
              greaters(x,nil()) -> nil()
              lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
              lowers(x,nil()) -> nil()
          - Signature:
              {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
              ,c_4/0,c_5/3,c_6/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
              ,nil}
*** Step 5.b:3.a:1: DecomposeDG MAYBE
    + Considered Problem:
        - Strict DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        and a lower component
          lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
        Further, following extension rules are added to the lower component.
          qsort#(.(x,y)) -> lowers#(x,y)
          qsort#(.(x,y)) -> qsort#(greaters(x,y))
          qsort#(.(x,y)) -> qsort#(lowers(x,y))
**** Step 5.b:3.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + 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: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
          
        The strictly oriented rules are moved into the weak component.
***** Step 5.b:3.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1,3}
        
        Following symbols are considered usable:
          {greaters,lowers,greaters#,lowers#,qsort#}
        TcT has computed the following interpretation:
                 p(++) = [1]                            
                  p(.) = [1] x1 + [12]                  
                 p(<=) = [1]                            
           p(greaters) = [0]                            
                 p(if) = [0]                            
             p(lowers) = [0]                            
                p(nil) = [0]                            
              p(qsort) = [2] x1 + [1]                   
          p(greaters#) = [2]                            
            p(lowers#) = [1] x1 + [0]                   
             p(qsort#) = [1] x1 + [0]                   
                p(c_1) = [2] x1 + [1] x2 + [1]          
                p(c_2) = [0]                            
                p(c_3) = [1] x1 + [4]                   
                p(c_4) = [2]                            
                p(c_5) = [4] x1 + [1] x2 + [1] x3 + [11]
                p(c_6) = [1]                            
        
        Following rules are strictly oriented:
        qsort#(.(x,y)) = [1] x + [12]                                               
                       > [1] x + [11]                                               
                       = c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        
        
        Following rules are (at-least) weakly oriented:
        greaters(x,.(y,z)) =  [0]                                         
                           >= [0]                                         
                           =  if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
        
         greaters(x,nil()) =  [0]                                         
                           >= [0]                                         
                           =  nil()                                       
        
          lowers(x,.(y,z)) =  [0]                                         
                           >= [0]                                         
                           =  if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))    
        
           lowers(x,nil()) =  [0]                                         
                           >= [0]                                         
                           =  nil()                                       
        
***** Step 5.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 5.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
***** Step 5.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 5.b:3.a:1.b:1: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
      - Weak DPs:
          qsort#(.(x,y)) -> lowers#(x,y)
          qsort#(.(x,y)) -> qsort#(greaters(x,y))
          qsort#(.(x,y)) -> qsort#(lowers(x,y))
      - Weak TRS:
          greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
          greaters(x,nil()) -> nil()
          lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
          lowers(x,nil()) -> nil()
      - Signature:
          {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
          ,c_4/0,c_5/3,c_6/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
          ,nil}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak DPs:
            lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
          
          2:W:lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
             -->_2 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
             -->_1 lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: lowers#(x,.(y,z)) -> c_3(lowers#(x,z),lowers#(x,z))
*** Step 5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/3,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y)))
             -->_3 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),lowers#(x,y),qsort#(greaters(x,y))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
*** Step 5.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_5) = {1,2}
        
        Following symbols are considered usable:
          {greaters,lowers,greaters#,lowers#,qsort#}
        TcT has computed the following interpretation:
                 p(++) = [1]                  
                  p(.) = [1]                  
                 p(<=) = [2]                  
           p(greaters) = [0]                  
                 p(if) = [0]                  
             p(lowers) = [0]                  
                p(nil) = [0]                  
              p(qsort) = [1]                  
          p(greaters#) = [2] x1 + [2]         
            p(lowers#) = [8] x2 + [4]         
             p(qsort#) = [1] x1 + [0]         
                p(c_1) = [1]                  
                p(c_2) = [2]                  
                p(c_3) = [2]                  
                p(c_4) = [0]                  
                p(c_5) = [4] x1 + [8] x2 + [0]
                p(c_6) = [1]                  
        
        Following rules are strictly oriented:
        qsort#(.(x,y)) = [1]                                           
                       > [0]                                           
                       = c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
        
        
        Following rules are (at-least) weakly oriented:
        greaters(x,.(y,z)) =  [0]                                         
                           >= [0]                                         
                           =  if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
        
         greaters(x,nil()) =  [0]                                         
                           >= [0]                                         
                           =  nil()                                       
        
          lowers(x,.(y,z)) =  [0]                                         
                           >= [0]                                         
                           =  if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))    
        
           lowers(x,nil()) =  [0]                                         
                           >= [0]                                         
                           =  nil()                                       
        
**** Step 5.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
             -->_2 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y))):1
             -->_1 qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: qsort#(.(x,y)) -> c_5(qsort#(lowers(x,y)),qsort#(greaters(x,y)))
**** Step 5.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
            greaters(x,nil()) -> nil()
            lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
            lowers(x,nil()) -> nil()
        - Signature:
            {greaters/2,lowers/2,qsort/1,greaters#/2,lowers#/2,qsort#/1} / {++/2,./2,<=/2,if/3,nil/0,c_1/2,c_2/0,c_3/2
            ,c_4/0,c_5/2,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {greaters#,lowers#,qsort#} and constructors {++,.,<=,if
            ,nil}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

MAYBE