MAYBE

Problem:
 qsort(nil()) -> nil()
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 lowers(x,nil()) -> nil()
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 greaters(x,nil()) -> nil()
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))

Proof:
 DP Processor:
  DPs:
   qsort#(.(x,y)) -> greaters#(x,y)
   qsort#(.(x,y)) -> qsort#(greaters(x,y))
   qsort#(.(x,y)) -> lowers#(x,y)
   qsort#(.(x,y)) -> qsort#(lowers(x,y))
   lowers#(x,.(y,z)) -> lowers#(x,z)
   greaters#(x,.(y,z)) -> greaters#(x,z)
  TRS:
   qsort(nil()) -> nil()
   qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
   lowers(x,nil()) -> nil()
   lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
   greaters(x,nil()) -> nil()
   greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
  TDG Processor:
   DPs:
    qsort#(.(x,y)) -> greaters#(x,y)
    qsort#(.(x,y)) -> qsort#(greaters(x,y))
    qsort#(.(x,y)) -> lowers#(x,y)
    qsort#(.(x,y)) -> qsort#(lowers(x,y))
    lowers#(x,.(y,z)) -> lowers#(x,z)
    greaters#(x,.(y,z)) -> greaters#(x,z)
   TRS:
    qsort(nil()) -> nil()
    qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
    lowers(x,nil()) -> nil()
    lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
    greaters(x,nil()) -> nil()
    greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
   graph:
    lowers#(x,.(y,z)) -> lowers#(x,z) ->
    lowers#(x,.(y,z)) -> lowers#(x,z)
    greaters#(x,.(y,z)) -> greaters#(x,z) ->
    greaters#(x,.(y,z)) -> greaters#(x,z)
    qsort#(.(x,y)) -> lowers#(x,y) ->
    lowers#(x,.(y,z)) -> lowers#(x,z)
    qsort#(.(x,y)) -> greaters#(x,y) ->
    greaters#(x,.(y,z)) -> greaters#(x,z)
    qsort#(.(x,y)) -> qsort#(greaters(x,y)) ->
    qsort#(.(x,y)) -> qsort#(lowers(x,y))
    qsort#(.(x,y)) -> qsort#(greaters(x,y)) ->
    qsort#(.(x,y)) -> lowers#(x,y)
    qsort#(.(x,y)) -> qsort#(greaters(x,y)) ->
    qsort#(.(x,y)) -> qsort#(greaters(x,y))
    qsort#(.(x,y)) -> qsort#(greaters(x,y)) ->
    qsort#(.(x,y)) -> greaters#(x,y)
    qsort#(.(x,y)) -> qsort#(lowers(x,y)) ->
    qsort#(.(x,y)) -> qsort#(lowers(x,y))
    qsort#(.(x,y)) -> qsort#(lowers(x,y)) ->
    qsort#(.(x,y)) -> lowers#(x,y)
    qsort#(.(x,y)) -> qsort#(lowers(x,y)) ->
    qsort#(.(x,y)) -> qsort#(greaters(x,y))
    qsort#(.(x,y)) -> qsort#(lowers(x,y)) -> qsort#(.(x,y)) -> greaters#(x,y)
   EDG Processor:
    DPs:
     qsort#(.(x,y)) -> greaters#(x,y)
     qsort#(.(x,y)) -> qsort#(greaters(x,y))
     qsort#(.(x,y)) -> lowers#(x,y)
     qsort#(.(x,y)) -> qsort#(lowers(x,y))
     lowers#(x,.(y,z)) -> lowers#(x,z)
     greaters#(x,.(y,z)) -> greaters#(x,z)
    TRS:
     qsort(nil()) -> nil()
     qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
     lowers(x,nil()) -> nil()
     lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
     greaters(x,nil()) -> nil()
     greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
    graph:
     lowers#(x,.(y,z)) -> lowers#(x,z) ->
     lowers#(x,.(y,z)) -> lowers#(x,z)
     greaters#(x,.(y,z)) -> greaters#(x,z) ->
     greaters#(x,.(y,z)) -> greaters#(x,z)
     qsort#(.(x,y)) -> lowers#(x,y) ->
     lowers#(x,.(y,z)) -> lowers#(x,z)
     qsort#(.(x,y)) -> greaters#(x,y) -> greaters#(x,.(y,z)) -> greaters#(x,z)
    SCC Processor:
     #sccs: 2
     #rules: 2
     #arcs: 4/36
     DPs:
      greaters#(x,.(y,z)) -> greaters#(x,z)
     TRS:
      qsort(nil()) -> nil()
      qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
      lowers(x,nil()) -> nil()
      lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
      greaters(x,nil()) -> nil()
      greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
     Open
     
     DPs:
      lowers#(x,.(y,z)) -> lowers#(x,z)
     TRS:
      qsort(nil()) -> nil()
      qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
      lowers(x,nil()) -> nil()
      lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
      greaters(x,nil()) -> nil()
      greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
     Open