(* If u,v is an enumeration of R, this sequence of open intervals
   tears the segment [0,1]. The recursive definition needs the proof that the
   previous interval is ordered, hence the type.

   The first sequence is increasing, the second decreasing.
   The first is below the second.
   Therefore the first sequence has a limit, a least upper bound b, that u cannot reach,
   which contradicts u (v b) = b. *)
Definition tearing_sequences (u : nat -> R) (v : R -> nat)
  : (enumeration R u v) -> nat -> { ab : prod R R | Rlt (fst ab) (snd ab) }.

Lemma tearing_sequences_inc_dec (u : nat -> R) (v : R -> nat) (pen : enumeration R u v) : forall n : nat, let I := proj1_sig (tearing_sequences u v pen n) in let SI := proj1_sig (tearing_sequences u v pen (S n)) in Rlt (fst I) (fst SI) /\ Rlt (snd SI) (snd I).