(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Definition enumeration (A : Type) (u : nat -> A) (v : A -> nat) : Prop :=
(forall x : A, u (v x) = x) /\ (forall n : nat, v (u n) = n).
Definition in_holed_interval (a b hole : R) (u : nat -> R) (n : nat) : Prop :=
Rlt a (u n) /\ Rlt (u n) b /\ u n <> hole.
Definition point_in_holed_interval (a b h : R) : R :=
if Req_EM_T h (Rdiv (Rplus a b) (INR 2)) then (Rdiv (Rplus a h) (INR 2))
else (Rdiv (Rplus a b) (INR 2)).
(* An enumeration of R reaches any open interval of R, extract the first two real numbers in it. *)
Definition first_in_holed_interval (u : nat -> R) (v : R -> nat) (a b h : R)
: enumeration R u v -> Rlt a b
-> { n : nat | in_holed_interval a b h u n
/\ forall k : nat, k < n -> ~in_holed_interval a b h u k }.
Definition first_two_in_interval (u : nat -> R) (v : R -> nat) (a b : R)
(pen : enumeration R u v) (plow : Rlt a b)
: prod R R :=
let (first_index, pr) := first_in_holed_interval u v a b b pen plow in
let (second_index, pr2) := first_in_holed_interval u v a b (u first_index) pen plow in
if Rle_dec (u first_index) (u second_index) then (u first_index, u second_index)
else (u second_index, u first_index).
(* 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) }.

(* The first tearing sequence in increasing, the second decreasing. That means the tearing sequences are nested intervals. *)
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).