Library CoqApprox.interval_compl

This file is part of the CoqApprox formalization of rigorous polynomial approximation in Coq:
Copyright (c) 2010-2013, ENS de Lyon and Inria.
This library is governed by the CeCILL-C license under French law and abiding by the rules of distribution of free software. You can use, modify and/or redistribute the library under the terms of the CeCILL-C license as circulated by CEA, CNRS and Inria at the following URL:
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Require Import Reals Psatz.
Require Import Interval_xreal.
Require Import Interval_interval.
Require Import ssreflect.
Require Import xreal_ssr_compat.

Set Implicit Arguments.

Lemma Xneg_as_Xmul (x : ExtendedR) : Xneg x = Xmul x (Xreal (-1)).

Lemma contains_trans (X : interval) (a b c : ExtendedR) :
  contains X a -> contains X b -> contains (Interval_interval.Ibnd a b) c ->
  contains X c.

Notation IIbnd := Interval_interval.Ibnd (only parsing).
Notation IInan := Interval_interval.Inan (only parsing).

Local Notation subset_ := Interval_interval.subset (only parsing).

Lemma subset_refl : forall x, subset_ x x.

Lemma contains_subset (X Y : interval) :
  (exists t, contains X t) ->
  (forall v : ExtendedR, contains X v -> contains Y v) ->
  subset_ X Y.