(************************************************************************)
(*         *   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)         *)
(************************************************************************)
Local Open Scope R_scope.
Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
match N with
| O => x
| S n =>
let down := Dichotomy_lb x y P n in
let up := Dichotomy_ub x y P n in
let z := (down + up) / 2 in if P z then down else z
end
with Dichotomy_ub (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
match N with
| O => y
| S n =>
let down := Dichotomy_lb x y P n in
let up := Dichotomy_ub x y P n in
let z := (down + up) / 2 in if P z then z else up
end.
Definition dicho_lb (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_lb x y P N.
Definition dicho_up (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_ub x y P N.
Definition pow_2_n (n:nat) := 2 ^ n.
Definition cond_positivity (x:R) : bool :=
match Rle_dec 0 x with
| left _ => true
| right _ => false
end.
intros a; unfold Rdiv; replace 0 with (a * 0) by ring.
apply CV_mult.
intros eps ep; exists 0%nat; rewrite R_dist_eq; intros n _; assumption.
exact (cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty).
Qed.

(** Intermediate Value Theorem *)
Lemma IVT :
forall (f:R -> R) (x y:R),
continuity f ->
x < y -> f x < 0 -> 0 < f y -> { z:R | x <= z <= y /\ f z = 0 }.