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))
  TDG 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))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X))
    top#(ok(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(head(X)) -> head#(active(X))
    top#(ok(X)) -> active#(X) -> active#(head(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X))
    top#(ok(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
    top#(ok(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X)
    top#(ok(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> incr#(XS)
    top#(ok(X)) -> active#(X) -> active#(odds()) -> incr#(pairs())
    top#(ok(X)) -> active#(X) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    top#(ok(X)) -> active#(X) -> active#(pairs()) -> incr#(odds())
    top#(ok(X)) -> active#(X) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    top#(ok(X)) -> active#(X) -> active#(nats()) -> incr#(nats())
    top#(mark(X)) -> top#(proper(X)) ->
    top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(tail(X)) -> proper#(X) ->
    proper#(tail(X)) -> tail#(proper(X))
    proper#(tail(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X)
    proper#(tail(X)) -> proper#(X) ->
    proper#(head(X)) -> head#(proper(X))
    proper#(tail(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X)
    proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(tail(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(tail(X)) -> proper#(X) ->
    proper#(incr(X)) -> incr#(proper(X))
    proper#(tail(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
    proper#(tail(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(tail(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(tail(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(tail(X)) -> tail#(proper(X)) ->
    tail#(ok(X)) -> tail#(X)
    proper#(tail(X)) -> tail#(proper(X)) ->
    tail#(mark(X)) -> tail#(X)
    proper#(head(X)) -> proper#(X) ->
    proper#(tail(X)) -> tail#(proper(X))
    proper#(head(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X)
    proper#(head(X)) -> proper#(X) ->
    proper#(head(X)) -> head#(proper(X))
    proper#(head(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X)
    proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(head(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(head(X)) -> proper#(X) ->
    proper#(incr(X)) -> incr#(proper(X))
    proper#(head(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
    proper#(head(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(head(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(head(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(head(X)) -> head#(proper(X)) ->
    head#(ok(X)) -> head#(X)
    proper#(head(X)) -> head#(proper(X)) -> head#(mark(X)) -> head#(X)
    proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> tail#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> head#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> incr#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X)
    proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(tail(X)) -> tail#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(tail(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(head(X)) -> head#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(head(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(incr(X)) -> incr#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(incr(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(tail(X)) -> tail#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(tail(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(head(X)) -> head#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(head(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(incr(X)) -> incr#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(incr(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    proper#(incr(X)) -> proper#(X) ->
    proper#(tail(X)) -> tail#(proper(X))
    proper#(incr(X)) -> proper#(X) -> proper#(tail(X)) -> proper#(X)
    proper#(incr(X)) -> proper#(X) ->
    proper#(head(X)) -> head#(proper(X))
    proper#(incr(X)) -> proper#(X) -> proper#(head(X)) -> proper#(X)
    proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(incr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(incr(X)) -> proper#(X) ->
    proper#(incr(X)) -> incr#(proper(X))
    proper#(incr(X)) -> proper#(X) -> proper#(incr(X)) -> proper#(X)
    proper#(incr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(incr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(incr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(incr(X)) -> incr#(proper(X)) ->
    incr#(ok(X)) -> incr#(X)
    proper#(incr(X)) -> incr#(proper(X)) -> incr#(mark(X)) -> incr#(X)
    tail#(ok(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X)
    tail#(ok(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X)
    tail#(mark(X)) -> tail#(X) -> tail#(ok(X)) -> tail#(X)
    tail#(mark(X)) -> tail#(X) -> tail#(mark(X)) -> tail#(X)
    head#(ok(X)) -> head#(X) -> head#(ok(X)) -> head#(X)
    head#(ok(X)) -> head#(X) -> head#(mark(X)) -> head#(X)
    head#(mark(X)) -> head#(X) -> head#(ok(X)) -> head#(X)
    head#(mark(X)) -> head#(X) -> head#(mark(X)) -> head#(X)
    s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    incr#(ok(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X)
    incr#(ok(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X)
    incr#(mark(X)) -> incr#(X) -> incr#(ok(X)) -> incr#(X)
    incr#(mark(X)) -> incr#(X) -> incr#(mark(X)) -> incr#(X)
    active#(tail(X)) -> tail#(active(X)) ->
    tail#(ok(X)) -> tail#(X)
    active#(tail(X)) -> tail#(active(X)) ->
    tail#(mark(X)) -> tail#(X)
    active#(tail(X)) -> active#(X) ->
    active#(tail(X)) -> tail#(active(X))
    active#(tail(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
    active#(tail(X)) -> active#(X) ->
    active#(head(X)) -> head#(active(X))
    active#(tail(X)) -> active#(X) -> active#(head(X)) -> active#(X)
    active#(tail(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(tail(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(tail(X)) -> active#(X) ->
    active#(incr(X)) -> incr#(active(X))
    active#(tail(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
    active#(tail(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(tail(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(tail(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    active#(tail(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> s#(X)
    active#(tail(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> incr#(XS)
    active#(tail(X)) -> active#(X) ->
    active#(odds()) -> incr#(pairs())
    active#(tail(X)) -> active#(X) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(tail(X)) -> active#(X) ->
    active#(pairs()) -> incr#(odds())
    active#(tail(X)) -> active#(X) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(tail(X)) -> active#(X) ->
    active#(nats()) -> incr#(nats())
    active#(head(X)) -> head#(active(X)) ->
    head#(ok(X)) -> head#(X)
    active#(head(X)) -> head#(active(X)) ->
    head#(mark(X)) -> head#(X)
    active#(head(X)) -> active#(X) ->
    active#(tail(X)) -> tail#(active(X))
    active#(head(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
    active#(head(X)) -> active#(X) ->
    active#(head(X)) -> head#(active(X))
    active#(head(X)) -> active#(X) -> active#(head(X)) -> active#(X)
    active#(head(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(head(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(head(X)) -> active#(X) ->
    active#(incr(X)) -> incr#(active(X))
    active#(head(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
    active#(head(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(head(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(head(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    active#(head(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> s#(X)
    active#(head(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> incr#(XS)
    active#(head(X)) -> active#(X) ->
    active#(odds()) -> incr#(pairs())
    active#(head(X)) -> active#(X) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(head(X)) -> active#(X) ->
    active#(pairs()) -> incr#(odds())
    active#(head(X)) -> active#(X) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(head(X)) -> active#(X) ->
    active#(nats()) -> incr#(nats())
    active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X)
    active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X)
    active#(s(X)) -> active#(X) -> active#(tail(X)) -> tail#(active(X))
    active#(s(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(head(X)) -> head#(active(X))
    active#(s(X)) -> active#(X) -> active#(head(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(incr(X)) -> incr#(active(X))
    active#(s(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    active#(s(X)) -> active#(X) -> active#(incr(cons(X,XS))) -> s#(X)
    active#(s(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> incr#(XS)
    active#(s(X)) -> active#(X) -> active#(odds()) -> incr#(pairs())
    active#(s(X)) -> active#(X) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(s(X)) -> active#(X) -> active#(pairs()) -> incr#(odds())
    active#(s(X)) -> active#(X) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(s(X)) -> active#(X) ->
    active#(nats()) -> incr#(nats())
    active#(odds()) -> incr#(pairs()) -> incr#(ok(X)) -> incr#(X)
    active#(odds()) -> incr#(pairs()) ->
    incr#(mark(X)) -> incr#(X)
    active#(pairs()) -> cons#(0(),incr(odds())) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(pairs()) -> cons#(0(),incr(odds())) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(pairs()) -> incr#(odds()) -> incr#(ok(X)) -> incr#(X)
    active#(pairs()) -> incr#(odds()) ->
    incr#(mark(X)) -> incr#(X)
    active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(tail(X)) -> tail#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(tail(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(head(X)) -> head#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(head(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(incr(X)) -> incr#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(incr(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),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))) -> incr#(XS)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(odds()) -> incr#(pairs())
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(pairs()) -> incr#(odds())
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(nats()) -> incr#(nats())
    active#(incr(cons(X,XS))) -> s#(X) -> s#(ok(X)) -> s#(X)
    active#(incr(cons(X,XS))) -> s#(X) ->
    s#(mark(X)) -> s#(X)
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS)) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(incr(cons(X,XS))) -> incr#(XS) ->
    incr#(ok(X)) -> incr#(X)
    active#(incr(cons(X,XS))) -> incr#(XS) ->
    incr#(mark(X)) -> incr#(X)
    active#(incr(X)) -> incr#(active(X)) ->
    incr#(ok(X)) -> incr#(X)
    active#(incr(X)) -> incr#(active(X)) ->
    incr#(mark(X)) -> incr#(X)
    active#(incr(X)) -> active#(X) ->
    active#(tail(X)) -> tail#(active(X))
    active#(incr(X)) -> active#(X) -> active#(tail(X)) -> active#(X)
    active#(incr(X)) -> active#(X) ->
    active#(head(X)) -> head#(active(X))
    active#(incr(X)) -> active#(X) -> active#(head(X)) -> active#(X)
    active#(incr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(incr(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(incr(X)) -> active#(X) ->
    active#(incr(X)) -> incr#(active(X))
    active#(incr(X)) -> active#(X) -> active#(incr(X)) -> active#(X)
    active#(incr(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(incr(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(incr(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> cons#(s(X),incr(XS))
    active#(incr(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> s#(X)
    active#(incr(X)) -> active#(X) ->
    active#(incr(cons(X,XS))) -> incr#(XS)
    active#(incr(X)) -> active#(X) ->
    active#(odds()) -> incr#(pairs())
    active#(incr(X)) -> active#(X) ->
    active#(pairs()) -> cons#(0(),incr(odds()))
    active#(incr(X)) -> active#(X) ->
    active#(pairs()) -> incr#(odds())
    active#(incr(X)) -> active#(X) ->
    active#(nats()) -> cons#(0(),incr(nats()))
    active#(incr(X)) -> active#(X) ->
    active#(nats()) -> incr#(nats())
    active#(nats()) -> cons#(0(),incr(nats())) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(nats()) -> cons#(0(),incr(nats())) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(nats()) -> incr#(nats()) -> incr#(ok(X)) -> incr#(X)
    active#(nats()) -> incr#(nats()) -> incr#(mark(X)) -> incr#(X)
   SCC Processor:
    #sccs: 8
    #rules: 23
    #arcs: 249/1849
    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#(cons(X1,X2)) -> active#(X1)
     active#(incr(X)) -> active#(X)
     active#(s(X)) -> active#(X)
     active#(head(X)) -> active#(X)
     active#(tail(X)) -> 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))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(incr(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X)
     proper#(head(X)) -> proper#(X)
     proper#(tail(X)) -> 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))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     tail#(mark(X)) -> tail#(X)
     tail#(ok(X)) -> tail#(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))
    Subterm Criterion Processor:
     simple projection:
      pi(tail#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     head#(mark(X)) -> head#(X)
     head#(ok(X)) -> head#(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))
    Subterm Criterion Processor:
     simple projection:
      pi(head#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     s#(mark(X)) -> s#(X)
     s#(ok(X)) -> s#(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))
    Subterm Criterion Processor:
     simple projection:
      pi(s#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     incr#(mark(X)) -> incr#(X)
     incr#(ok(X)) -> incr#(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))
    Subterm Criterion Processor:
     simple projection:
      pi(incr#) = 0
     problem:
      DPs:
       
      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))
     Qed
    
    DPs:
     cons#(mark(X1),X2) -> cons#(X1,X2)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    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))
    Subterm Criterion Processor:
     simple projection:
      pi(cons#) = 1
     problem:
      DPs:
       cons#(mark(X1),X2) -> cons#(X1,X2)
      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))
     Subterm Criterion Processor:
      simple projection:
       pi(cons#) = 0
      problem:
       DPs:
        
       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))
      Qed