MAYBE

Problem:
 pairNs() -> cons(0(),n__incr(oddNs()))
 oddNs() -> incr(pairNs())
 incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
 take(0(),XS) -> nil()
 take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
 zip(nil(),XS) -> nil()
 zip(X,nil()) -> nil()
 zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS)))
 tail(cons(X,XS)) -> activate(XS)
 repItems(nil()) -> nil()
 repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS))))
 incr(X) -> n__incr(X)
 take(X1,X2) -> n__take(X1,X2)
 zip(X1,X2) -> n__zip(X1,X2)
 cons(X1,X2) -> n__cons(X1,X2)
 repItems(X) -> n__repItems(X)
 activate(n__incr(X)) -> incr(X)
 activate(n__take(X1,X2)) -> take(X1,X2)
 activate(n__zip(X1,X2)) -> zip(X1,X2)
 activate(n__cons(X1,X2)) -> cons(X1,X2)
 activate(n__repItems(X)) -> repItems(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   pairNs#() -> oddNs#()
   pairNs#() -> cons#(0(),n__incr(oddNs()))
   oddNs#() -> pairNs#()
   oddNs#() -> incr#(pairNs())
   incr#(cons(X,XS)) -> activate#(XS)
   incr#(cons(X,XS)) -> cons#(s(X),n__incr(activate(XS)))
   take#(s(N),cons(X,XS)) -> activate#(XS)
   take#(s(N),cons(X,XS)) -> cons#(X,n__take(N,activate(XS)))
   zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS)
   zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS)
   zip#(cons(X,XS),cons(Y,YS)) -> cons#(pair(X,Y),n__zip(activate(XS),activate(YS)))
   tail#(cons(X,XS)) -> activate#(XS)
   repItems#(cons(X,XS)) -> activate#(XS)
   repItems#(cons(X,XS)) -> cons#(X,n__cons(X,n__repItems(activate(XS))))
   activate#(n__incr(X)) -> incr#(X)
   activate#(n__take(X1,X2)) -> take#(X1,X2)
   activate#(n__zip(X1,X2)) -> zip#(X1,X2)
   activate#(n__cons(X1,X2)) -> cons#(X1,X2)
   activate#(n__repItems(X)) -> repItems#(X)
  TRS:
   pairNs() -> cons(0(),n__incr(oddNs()))
   oddNs() -> incr(pairNs())
   incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
   take(0(),XS) -> nil()
   take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
   zip(nil(),XS) -> nil()
   zip(X,nil()) -> nil()
   zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS)))
   tail(cons(X,XS)) -> activate(XS)
   repItems(nil()) -> nil()
   repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS))))
   incr(X) -> n__incr(X)
   take(X1,X2) -> n__take(X1,X2)
   zip(X1,X2) -> n__zip(X1,X2)
   cons(X1,X2) -> n__cons(X1,X2)
   repItems(X) -> n__repItems(X)
   activate(n__incr(X)) -> incr(X)
   activate(n__take(X1,X2)) -> take(X1,X2)
   activate(n__zip(X1,X2)) -> zip(X1,X2)
   activate(n__cons(X1,X2)) -> cons(X1,X2)
   activate(n__repItems(X)) -> repItems(X)
   activate(X) -> X
  ADG Processor:
   DPs:
    pairNs#() -> oddNs#()
    pairNs#() -> cons#(0(),n__incr(oddNs()))
    oddNs#() -> pairNs#()
    oddNs#() -> incr#(pairNs())
    incr#(cons(X,XS)) -> activate#(XS)
    incr#(cons(X,XS)) -> cons#(s(X),n__incr(activate(XS)))
    take#(s(N),cons(X,XS)) -> activate#(XS)
    take#(s(N),cons(X,XS)) -> cons#(X,n__take(N,activate(XS)))
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS)
    zip#(cons(X,XS),cons(Y,YS)) -> cons#(pair(X,Y),n__zip(activate(XS),activate(YS)))
    tail#(cons(X,XS)) -> activate#(XS)
    repItems#(cons(X,XS)) -> activate#(XS)
    repItems#(cons(X,XS)) -> cons#(X,n__cons(X,n__repItems(activate(XS))))
    activate#(n__incr(X)) -> incr#(X)
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    activate#(n__repItems(X)) -> repItems#(X)
   TRS:
    pairNs() -> cons(0(),n__incr(oddNs()))
    oddNs() -> incr(pairNs())
    incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
    take(0(),XS) -> nil()
    take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
    zip(nil(),XS) -> nil()
    zip(X,nil()) -> nil()
    zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS)))
    tail(cons(X,XS)) -> activate(XS)
    repItems(nil()) -> nil()
    repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS))))
    incr(X) -> n__incr(X)
    take(X1,X2) -> n__take(X1,X2)
    zip(X1,X2) -> n__zip(X1,X2)
    cons(X1,X2) -> n__cons(X1,X2)
    repItems(X) -> n__repItems(X)
    activate(n__incr(X)) -> incr(X)
    activate(n__take(X1,X2)) -> take(X1,X2)
    activate(n__zip(X1,X2)) -> zip(X1,X2)
    activate(n__cons(X1,X2)) -> cons(X1,X2)
    activate(n__repItems(X)) -> repItems(X)
    activate(X) -> X
   graph:
    repItems#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__incr(X)) -> incr#(X)
    repItems#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    repItems#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    repItems#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    repItems#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    tail#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__incr(X)) -> incr#(X)
    tail#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    tail#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    tail#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    tail#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS) ->
    activate#(n__incr(X)) -> incr#(X)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS) ->
    activate#(n__incr(X)) -> incr#(X)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    take#(s(N),cons(X,XS)) -> activate#(XS) ->
    activate#(n__incr(X)) -> incr#(X)
    take#(s(N),cons(X,XS)) -> activate#(XS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    take#(s(N),cons(X,XS)) -> activate#(XS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    take#(s(N),cons(X,XS)) -> activate#(XS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    take#(s(N),cons(X,XS)) -> activate#(XS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    activate#(n__repItems(X)) -> repItems#(X) ->
    repItems#(cons(X,XS)) -> activate#(XS)
    activate#(n__repItems(X)) -> repItems#(X) ->
    repItems#(cons(X,XS)) -> cons#(X,n__cons(X,n__repItems(activate(XS))))
    activate#(n__zip(X1,X2)) -> zip#(X1,X2) ->
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS)
    activate#(n__zip(X1,X2)) -> zip#(X1,X2) ->
    zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS)
    activate#(n__zip(X1,X2)) -> zip#(X1,X2) ->
    zip#(cons(X,XS),cons(Y,YS)) -> cons#(pair(X,Y),n__zip(activate(XS),activate(YS)))
    activate#(n__take(X1,X2)) -> take#(X1,X2) ->
    take#(s(N),cons(X,XS)) -> activate#(XS)
    activate#(n__take(X1,X2)) -> take#(X1,X2) ->
    take#(s(N),cons(X,XS)) -> cons#(X,n__take(N,activate(XS)))
    activate#(n__incr(X)) -> incr#(X) ->
    incr#(cons(X,XS)) -> activate#(XS)
    activate#(n__incr(X)) -> incr#(X) ->
    incr#(cons(X,XS)) -> cons#(s(X),n__incr(activate(XS)))
    incr#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__incr(X)) -> incr#(X)
    incr#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__take(X1,X2)) -> take#(X1,X2)
    incr#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__zip(X1,X2)) -> zip#(X1,X2)
    incr#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__cons(X1,X2)) -> cons#(X1,X2)
    incr#(cons(X,XS)) -> activate#(XS) ->
    activate#(n__repItems(X)) -> repItems#(X)
    oddNs#() -> incr#(pairNs()) -> incr#(cons(X,XS)) -> activate#(XS)
    oddNs#() -> incr#(pairNs()) ->
    incr#(cons(X,XS)) -> cons#(s(X),n__incr(activate(XS)))
    oddNs#() -> pairNs#() -> pairNs#() -> oddNs#()
    oddNs#() -> pairNs#() -> pairNs#() -> cons#(0(),n__incr(oddNs()))
    pairNs#() -> oddNs#() -> oddNs#() -> pairNs#()
    pairNs#() -> oddNs#() -> oddNs#() -> incr#(pairNs())
   SCC Processor:
    #sccs: 2
    #rules: 11
    #arcs: 45/361
    DPs:
     oddNs#() -> pairNs#()
     pairNs#() -> oddNs#()
    TRS:
     pairNs() -> cons(0(),n__incr(oddNs()))
     oddNs() -> incr(pairNs())
     incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
     take(0(),XS) -> nil()
     take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
     zip(nil(),XS) -> nil()
     zip(X,nil()) -> nil()
     zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS)))
     tail(cons(X,XS)) -> activate(XS)
     repItems(nil()) -> nil()
     repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS))))
     incr(X) -> n__incr(X)
     take(X1,X2) -> n__take(X1,X2)
     zip(X1,X2) -> n__zip(X1,X2)
     cons(X1,X2) -> n__cons(X1,X2)
     repItems(X) -> n__repItems(X)
     activate(n__incr(X)) -> incr(X)
     activate(n__take(X1,X2)) -> take(X1,X2)
     activate(n__zip(X1,X2)) -> zip(X1,X2)
     activate(n__cons(X1,X2)) -> cons(X1,X2)
     activate(n__repItems(X)) -> repItems(X)
     activate(X) -> X
    Open
    
    DPs:
     repItems#(cons(X,XS)) -> activate#(XS)
     activate#(n__repItems(X)) -> repItems#(X)
     activate#(n__zip(X1,X2)) -> zip#(X1,X2)
     zip#(cons(X,XS),cons(Y,YS)) -> activate#(XS)
     activate#(n__take(X1,X2)) -> take#(X1,X2)
     take#(s(N),cons(X,XS)) -> activate#(XS)
     activate#(n__incr(X)) -> incr#(X)
     incr#(cons(X,XS)) -> activate#(XS)
     zip#(cons(X,XS),cons(Y,YS)) -> activate#(YS)
    TRS:
     pairNs() -> cons(0(),n__incr(oddNs()))
     oddNs() -> incr(pairNs())
     incr(cons(X,XS)) -> cons(s(X),n__incr(activate(XS)))
     take(0(),XS) -> nil()
     take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS)))
     zip(nil(),XS) -> nil()
     zip(X,nil()) -> nil()
     zip(cons(X,XS),cons(Y,YS)) -> cons(pair(X,Y),n__zip(activate(XS),activate(YS)))
     tail(cons(X,XS)) -> activate(XS)
     repItems(nil()) -> nil()
     repItems(cons(X,XS)) -> cons(X,n__cons(X,n__repItems(activate(XS))))
     incr(X) -> n__incr(X)
     take(X1,X2) -> n__take(X1,X2)
     zip(X1,X2) -> n__zip(X1,X2)
     cons(X1,X2) -> n__cons(X1,X2)
     repItems(X) -> n__repItems(X)
     activate(n__incr(X)) -> incr(X)
     activate(n__take(X1,X2)) -> take(X1,X2)
     activate(n__zip(X1,X2)) -> zip(X1,X2)
     activate(n__cons(X1,X2)) -> cons(X1,X2)
     activate(n__repItems(X)) -> repItems(X)
     activate(X) -> X
    Open