MAYBE

Problem:
 active(nats()) -> mark(cons(0(),incr(nats())))
 active(pairs()) -> mark(cons(0(),incr(odds())))
 active(odds()) -> mark(incr(pairs()))
 active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
 active(head(cons(X,XS))) -> mark(X)
 active(tail(cons(X,XS))) -> mark(XS)
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(incr(X)) -> incr(active(X))
 active(s(X)) -> s(active(X))
 active(head(X)) -> head(active(X))
 active(tail(X)) -> tail(active(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 incr(mark(X)) -> mark(incr(X))
 s(mark(X)) -> mark(s(X))
 head(mark(X)) -> mark(head(X))
 tail(mark(X)) -> mark(tail(X))
 proper(nats()) -> ok(nats())
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(incr(X)) -> incr(proper(X))
 proper(pairs()) -> ok(pairs())
 proper(odds()) -> ok(odds())
 proper(s(X)) -> s(proper(X))
 proper(head(X)) -> head(proper(X))
 proper(tail(X)) -> tail(proper(X))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 incr(ok(X)) -> ok(incr(X))
 s(ok(X)) -> ok(s(X))
 head(ok(X)) -> ok(head(X))
 tail(ok(X)) -> ok(tail(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(nats()) -> incr#(nats())
   active#(nats()) -> cons#(0(),incr(nats()))
   active#(pairs()) -> incr#(odds())
   active#(pairs()) -> cons#(0(),incr(odds()))
   active#(odds()) -> incr#(pairs())
   active#(incr(cons(X,XS))) -> incr#(XS)
   active#(incr(cons(X,XS))) -> s#(X)
   active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
   active#(cons(X1,X2)) -> active#(X1)
   active#(cons(X1,X2)) -> cons#(active(X1),X2)
   active#(incr(X)) -> active#(X)
   active#(incr(X)) -> incr#(active(X))
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(head(X)) -> active#(X)
   active#(head(X)) -> head#(active(X))
   active#(tail(X)) -> active#(X)
   active#(tail(X)) -> tail#(active(X))
   cons#(mark(X1),X2) -> cons#(X1,X2)
   incr#(mark(X)) -> incr#(X)
   s#(mark(X)) -> s#(X)
   head#(mark(X)) -> head#(X)
   tail#(mark(X)) -> tail#(X)
   proper#(cons(X1,X2)) -> proper#(X2)
   proper#(cons(X1,X2)) -> proper#(X1)
   proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
   proper#(incr(X)) -> proper#(X)
   proper#(incr(X)) -> incr#(proper(X))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(head(X)) -> proper#(X)
   proper#(head(X)) -> head#(proper(X))
   proper#(tail(X)) -> proper#(X)
   proper#(tail(X)) -> tail#(proper(X))
   cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
   incr#(ok(X)) -> incr#(X)
   s#(ok(X)) -> s#(X)
   head#(ok(X)) -> head#(X)
   tail#(ok(X)) -> tail#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(nats()) -> mark(cons(0(),incr(nats())))
   active(pairs()) -> mark(cons(0(),incr(odds())))
   active(odds()) -> mark(incr(pairs()))
   active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
   active(head(cons(X,XS))) -> mark(X)
   active(tail(cons(X,XS))) -> mark(XS)
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(incr(X)) -> incr(active(X))
   active(s(X)) -> s(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   incr(mark(X)) -> mark(incr(X))
   s(mark(X)) -> mark(s(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(nats()) -> ok(nats())
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(incr(X)) -> incr(proper(X))
   proper(pairs()) -> ok(pairs())
   proper(odds()) -> ok(odds())
   proper(s(X)) -> s(proper(X))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Usable Rule Processor:
   DPs:
    active#(nats()) -> incr#(nats())
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(pairs()) -> incr#(odds())
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(odds()) -> incr#(pairs())
    active#(incr(cons(X,XS))) -> incr#(XS)
    active#(incr(cons(X,XS))) -> s#(X)
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(incr(X)) -> active#(X)
    active#(incr(X)) -> incr#(active(X))
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(head(X)) -> active#(X)
    active#(head(X)) -> head#(active(X))
    active#(tail(X)) -> active#(X)
    active#(tail(X)) -> tail#(active(X))
    cons#(mark(X1),X2) -> cons#(X1,X2)
    incr#(mark(X)) -> incr#(X)
    s#(mark(X)) -> s#(X)
    head#(mark(X)) -> head#(X)
    tail#(mark(X)) -> tail#(X)
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(incr(X)) -> proper#(X)
    proper#(incr(X)) -> incr#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(head(X)) -> proper#(X)
    proper#(head(X)) -> head#(proper(X))
    proper#(tail(X)) -> proper#(X)
    proper#(tail(X)) -> tail#(proper(X))
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    incr#(ok(X)) -> incr#(X)
    s#(ok(X)) -> s#(X)
    head#(ok(X)) -> head#(X)
    tail#(ok(X)) -> tail#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    incr(mark(X)) -> mark(incr(X))
    incr(ok(X)) -> ok(incr(X))
    s(mark(X)) -> mark(s(X))
    s(ok(X)) -> ok(s(X))
    active(nats()) -> mark(cons(0(),incr(nats())))
    active(pairs()) -> mark(cons(0(),incr(odds())))
    active(odds()) -> mark(incr(pairs()))
    active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
    active(head(cons(X,XS))) -> mark(X)
    active(tail(cons(X,XS))) -> mark(XS)
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(incr(X)) -> incr(active(X))
    active(s(X)) -> s(active(X))
    active(head(X)) -> head(active(X))
    active(tail(X)) -> tail(active(X))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    head(mark(X)) -> mark(head(X))
    head(ok(X)) -> ok(head(X))
    tail(mark(X)) -> mark(tail(X))
    tail(ok(X)) -> ok(tail(X))
    proper(nats()) -> ok(nats())
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(incr(X)) -> incr(proper(X))
    proper(pairs()) -> ok(pairs())
    proper(odds()) -> ok(odds())
    proper(s(X)) -> s(proper(X))
    proper(head(X)) -> head(proper(X))
    proper(tail(X)) -> tail(proper(X))
   CDG Processor:
    DPs:
     active#(nats()) -> incr#(nats())
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(pairs()) -> incr#(odds())
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(odds()) -> incr#(pairs())
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(incr(cons(X,XS))) -> s#(X)
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(cons(X1,X2)) -> active#(X1)
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(incr(X)) -> active#(X)
     active#(incr(X)) -> incr#(active(X))
     active#(s(X)) -> active#(X)
     active#(s(X)) -> s#(active(X))
     active#(head(X)) -> active#(X)
     active#(head(X)) -> head#(active(X))
     active#(tail(X)) -> active#(X)
     active#(tail(X)) -> tail#(active(X))
     cons#(mark(X1),X2) -> cons#(X1,X2)
     incr#(mark(X)) -> incr#(X)
     s#(mark(X)) -> s#(X)
     head#(mark(X)) -> head#(X)
     tail#(mark(X)) -> tail#(X)
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
     proper#(incr(X)) -> proper#(X)
     proper#(incr(X)) -> incr#(proper(X))
     proper#(s(X)) -> proper#(X)
     proper#(s(X)) -> s#(proper(X))
     proper#(head(X)) -> proper#(X)
     proper#(head(X)) -> head#(proper(X))
     proper#(tail(X)) -> proper#(X)
     proper#(tail(X)) -> tail#(proper(X))
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     incr#(ok(X)) -> incr#(X)
     s#(ok(X)) -> s#(X)
     head#(ok(X)) -> head#(X)
     tail#(ok(X)) -> tail#(X)
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X))
    TRS:
     incr(mark(X)) -> mark(incr(X))
     incr(ok(X)) -> ok(incr(X))
     s(mark(X)) -> mark(s(X))
     s(ok(X)) -> ok(s(X))
     active(nats()) -> mark(cons(0(),incr(nats())))
     active(pairs()) -> mark(cons(0(),incr(odds())))
     active(odds()) -> mark(incr(pairs()))
     active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
     active(head(cons(X,XS))) -> mark(X)
     active(tail(cons(X,XS))) -> mark(XS)
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(incr(X)) -> incr(active(X))
     active(s(X)) -> s(active(X))
     active(head(X)) -> head(active(X))
     active(tail(X)) -> tail(active(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     head(mark(X)) -> mark(head(X))
     head(ok(X)) -> ok(head(X))
     tail(mark(X)) -> mark(tail(X))
     tail(ok(X)) -> ok(tail(X))
     proper(nats()) -> ok(nats())
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(0()) -> ok(0())
     proper(incr(X)) -> incr(proper(X))
     proper(pairs()) -> ok(pairs())
     proper(odds()) -> ok(odds())
     proper(s(X)) -> s(proper(X))
     proper(head(X)) -> head(proper(X))
     proper(tail(X)) -> tail(proper(X))
    graph:
     top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
     top#(ok(X)) -> top#(active(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
     top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
     top#(ok(X)) -> active#(X) -> active#(nats()) -> incr#(nats())
     top#(ok(X)) -> active#(X) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     top#(ok(X)) -> active#(X) -> active#(pairs()) -> incr#(odds())
     top#(ok(X)) -> active#(X) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     top#(ok(X)) -> active#(X) -> active#(odds()) -> incr#(pairs())
     top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS)
     top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X)
     top#(ok(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
     top#(ok(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X))
     top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
     top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X)
     top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X))
     top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
     top#(ok(X)) -> active#(X) ->
     active#(tail(X)) -> tail#(active(X))
     top#(mark(X)) -> top#(proper(X)) ->
     top#(mark(X)) -> proper#(X)
     top#(mark(X)) -> top#(proper(X)) ->
     top#(mark(X)) -> top#(proper(X))
     top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
     top#(mark(X)) -> top#(proper(X)) ->
     top#(ok(X)) -> top#(active(X))
     proper#(tail(X)) -> tail#(proper(X)) ->
     tail#(ok(X)) -> tail#(X)
     proper#(head(X)) -> head#(proper(X)) -> head#(ok(X)) -> head#(X)
     proper#(s(X)) -> s#(proper(X)) ->
     s#(ok(X)) -> s#(X)
     proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     proper#(incr(X)) -> incr#(proper(X)) ->
     incr#(ok(X)) -> incr#(X)
     active#(tail(X)) -> tail#(active(X)) ->
     tail#(mark(X)) -> tail#(X)
     active#(tail(X)) -> tail#(active(X)) -> tail#(ok(X)) -> tail#(X)
     active#(tail(X)) -> active#(X) ->
     active#(nats()) -> incr#(nats())
     active#(tail(X)) -> active#(X) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(tail(X)) -> active#(X) ->
     active#(pairs()) -> incr#(odds())
     active#(tail(X)) -> active#(X) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(tail(X)) -> active#(X) ->
     active#(odds()) -> incr#(pairs())
     active#(tail(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(tail(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> s#(X)
     active#(tail(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(tail(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> active#(X1)
     active#(tail(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(tail(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X)
     active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
     active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X)
     active#(tail(X)) -> active#(X) ->
     active#(head(X)) -> head#(active(X))
     active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
     active#(tail(X)) -> active#(X) ->
     active#(tail(X)) -> tail#(active(X))
     active#(head(X)) -> head#(active(X)) ->
     head#(mark(X)) -> head#(X)
     active#(head(X)) -> head#(active(X)) -> head#(ok(X)) -> head#(X)
     active#(head(X)) -> active#(X) ->
     active#(nats()) -> incr#(nats())
     active#(head(X)) -> active#(X) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(head(X)) -> active#(X) ->
     active#(pairs()) -> incr#(odds())
     active#(head(X)) -> active#(X) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(head(X)) -> active#(X) ->
     active#(odds()) -> incr#(pairs())
     active#(head(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(head(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> s#(X)
     active#(head(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(head(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> active#(X1)
     active#(head(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(head(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X)
     active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
     active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X)
     active#(head(X)) -> active#(X) ->
     active#(head(X)) -> head#(active(X))
     active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
     active#(head(X)) -> active#(X) ->
     active#(tail(X)) -> tail#(active(X))
     active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X)
     active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X)
     active#(s(X)) -> active#(X) -> active#(nats()) -> incr#(nats())
     active#(s(X)) -> active#(X) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(s(X)) -> active#(X) -> active#(pairs()) -> incr#(odds())
     active#(s(X)) -> active#(X) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(s(X)) -> active#(X) -> active#(odds()) -> incr#(pairs())
     active#(s(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X)
     active#(s(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
     active#(s(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X))
     active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
     active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
     active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X)
     active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X))
     active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
     active#(s(X)) -> active#(X) ->
     active#(tail(X)) -> tail#(active(X))
     active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
     cons#(mark(X1),X2) -> cons#(X1,X2)
     active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(nats()) -> incr#(nats())
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(pairs()) -> incr#(odds())
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(odds()) -> incr#(pairs())
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(incr(cons(X,XS))) -> s#(X)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(cons(X1,X2)) -> active#(X1)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(incr(X)) -> active#(X)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(incr(X)) -> incr#(active(X))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(s(X)) -> active#(X)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(s(X)) -> s#(active(X))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(head(X)) -> active#(X)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(head(X)) -> head#(active(X))
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(tail(X)) -> active#(X)
     active#(cons(X1,X2)) -> active#(X1) ->
     active#(tail(X)) -> tail#(active(X))
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) ->
     cons#(mark(X1),X2) -> cons#(X1,X2)
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) ->
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
     active#(incr(X)) -> incr#(active(X)) ->
     incr#(mark(X)) -> incr#(X)
     active#(incr(X)) -> incr#(active(X)) -> incr#(ok(X)) -> incr#(X)
     active#(incr(X)) -> active#(X) ->
     active#(nats()) -> incr#(nats())
     active#(incr(X)) -> active#(X) ->
     active#(nats()) -> cons#(0(),incr(nats()))
     active#(incr(X)) -> active#(X) ->
     active#(pairs()) -> incr#(odds())
     active#(incr(X)) -> active#(X) ->
     active#(pairs()) -> cons#(0(),incr(odds()))
     active#(incr(X)) -> active#(X) ->
     active#(odds()) -> incr#(pairs())
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> incr#(XS)
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> s#(X)
     active#(incr(X)) -> active#(X) ->
     active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
     active#(incr(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> active#(X1)
     active#(incr(X)) -> active#(X) ->
     active#(cons(X1,X2)) -> cons#(active(X1),X2)
     active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(incr(X)) -> incr#(active(X))
     active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X)
     active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
     active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X)
     active#(incr(X)) -> active#(X) ->
     active#(head(X)) -> head#(active(X))
     active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
     active#(incr(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X))
    Restore Modifier:
     DPs:
      active#(nats()) -> incr#(nats())
      active#(nats()) -> cons#(0(),incr(nats()))
      active#(pairs()) -> incr#(odds())
      active#(pairs()) -> cons#(0(),incr(odds()))
      active#(odds()) -> incr#(pairs())
      active#(incr(cons(X,XS))) -> incr#(XS)
      active#(incr(cons(X,XS))) -> s#(X)
      active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
      active#(cons(X1,X2)) -> active#(X1)
      active#(cons(X1,X2)) -> cons#(active(X1),X2)
      active#(incr(X)) -> active#(X)
      active#(incr(X)) -> incr#(active(X))
      active#(s(X)) -> active#(X)
      active#(s(X)) -> s#(active(X))
      active#(head(X)) -> active#(X)
      active#(head(X)) -> head#(active(X))
      active#(tail(X)) -> active#(X)
      active#(tail(X)) -> tail#(active(X))
      cons#(mark(X1),X2) -> cons#(X1,X2)
      incr#(mark(X)) -> incr#(X)
      s#(mark(X)) -> s#(X)
      head#(mark(X)) -> head#(X)
      tail#(mark(X)) -> tail#(X)
      proper#(cons(X1,X2)) -> proper#(X2)
      proper#(cons(X1,X2)) -> proper#(X1)
      proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
      proper#(incr(X)) -> proper#(X)
      proper#(incr(X)) -> incr#(proper(X))
      proper#(s(X)) -> proper#(X)
      proper#(s(X)) -> s#(proper(X))
      proper#(head(X)) -> proper#(X)
      proper#(head(X)) -> head#(proper(X))
      proper#(tail(X)) -> proper#(X)
      proper#(tail(X)) -> tail#(proper(X))
      cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
      incr#(ok(X)) -> incr#(X)
      s#(ok(X)) -> s#(X)
      head#(ok(X)) -> head#(X)
      tail#(ok(X)) -> tail#(X)
      top#(mark(X)) -> proper#(X)
      top#(mark(X)) -> top#(proper(X))
      top#(ok(X)) -> active#(X)
      top#(ok(X)) -> top#(active(X))
     TRS:
      active(nats()) -> mark(cons(0(),incr(nats())))
      active(pairs()) -> mark(cons(0(),incr(odds())))
      active(odds()) -> mark(incr(pairs()))
      active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
      active(head(cons(X,XS))) -> mark(X)
      active(tail(cons(X,XS))) -> mark(XS)
      active(cons(X1,X2)) -> cons(active(X1),X2)
      active(incr(X)) -> incr(active(X))
      active(s(X)) -> s(active(X))
      active(head(X)) -> head(active(X))
      active(tail(X)) -> tail(active(X))
      cons(mark(X1),X2) -> mark(cons(X1,X2))
      incr(mark(X)) -> mark(incr(X))
      s(mark(X)) -> mark(s(X))
      head(mark(X)) -> mark(head(X))
      tail(mark(X)) -> mark(tail(X))
      proper(nats()) -> ok(nats())
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(0()) -> ok(0())
      proper(incr(X)) -> incr(proper(X))
      proper(pairs()) -> ok(pairs())
      proper(odds()) -> ok(odds())
      proper(s(X)) -> s(proper(X))
      proper(head(X)) -> head(proper(X))
      proper(tail(X)) -> tail(proper(X))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      incr(ok(X)) -> ok(incr(X))
      s(ok(X)) -> ok(s(X))
      head(ok(X)) -> ok(head(X))
      tail(ok(X)) -> ok(tail(X))
      top(mark(X)) -> top(proper(X))
      top(ok(X)) -> top(active(X))
     SCC Processor:
      #sccs: 2
      #rules: 7
      #arcs: 133/1849
      DPs:
       top#(ok(X)) -> top#(active(X))
       top#(mark(X)) -> top#(proper(X))
      TRS:
       active(nats()) -> mark(cons(0(),incr(nats())))
       active(pairs()) -> mark(cons(0(),incr(odds())))
       active(odds()) -> mark(incr(pairs()))
       active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
       active(head(cons(X,XS))) -> mark(X)
       active(tail(cons(X,XS))) -> mark(XS)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(incr(X)) -> incr(active(X))
       active(s(X)) -> s(active(X))
       active(head(X)) -> head(active(X))
       active(tail(X)) -> tail(active(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       incr(mark(X)) -> mark(incr(X))
       s(mark(X)) -> mark(s(X))
       head(mark(X)) -> mark(head(X))
       tail(mark(X)) -> mark(tail(X))
       proper(nats()) -> ok(nats())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(incr(X)) -> incr(proper(X))
       proper(pairs()) -> ok(pairs())
       proper(odds()) -> ok(odds())
       proper(s(X)) -> s(proper(X))
       proper(head(X)) -> head(proper(X))
       proper(tail(X)) -> tail(proper(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       incr(ok(X)) -> ok(incr(X))
       s(ok(X)) -> ok(s(X))
       head(ok(X)) -> ok(head(X))
       tail(ok(X)) -> ok(tail(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open
      
      DPs:
       active#(tail(X)) -> active#(X)
       active#(head(X)) -> active#(X)
       active#(s(X)) -> active#(X)
       active#(incr(X)) -> active#(X)
       active#(cons(X1,X2)) -> active#(X1)
      TRS:
       active(nats()) -> mark(cons(0(),incr(nats())))
       active(pairs()) -> mark(cons(0(),incr(odds())))
       active(odds()) -> mark(incr(pairs()))
       active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS)))
       active(head(cons(X,XS))) -> mark(X)
       active(tail(cons(X,XS))) -> mark(XS)
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(incr(X)) -> incr(active(X))
       active(s(X)) -> s(active(X))
       active(head(X)) -> head(active(X))
       active(tail(X)) -> tail(active(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       incr(mark(X)) -> mark(incr(X))
       s(mark(X)) -> mark(s(X))
       head(mark(X)) -> mark(head(X))
       tail(mark(X)) -> mark(tail(X))
       proper(nats()) -> ok(nats())
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(0()) -> ok(0())
       proper(incr(X)) -> incr(proper(X))
       proper(pairs()) -> ok(pairs())
       proper(odds()) -> ok(odds())
       proper(s(X)) -> s(proper(X))
       proper(head(X)) -> head(proper(X))
       proper(tail(X)) -> tail(proper(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       incr(ok(X)) -> ok(incr(X))
       s(ok(X)) -> ok(s(X))
       head(ok(X)) -> ok(head(X))
       tail(ok(X)) -> ok(tail(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
      Open