(* Contribution to the Coq Library   V6.3 (July 1999)                       *)
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(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.11                                 *)
(*                              Feb 2nd 1996                                *)
(*                                                                          *)
(*                (notations and layout updated March 2009)                 *)
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(*                               Schroeder.v                                *)
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(* This file is distributed under the terms of the                          *)
(* GNU Lesser General Public License Version 2.1                            *)
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(**  If A is of cardinal less than B and conversely, then A and B           *)
(**  are equipollent                                                        *)
(**  In other words, if there is an injective map from A to B and           *)
(**  an injective map from B to A then there exists a map from A onto B.    *)
(**                  (based on a proof by Fraenkel)                         *)
Set Nested Proofs Allowed.
Section Schroeder_Bernstein.
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(** We need the decidability of the belonging relation on sets              *)
(** This is equivalent to classical logic                                   *)
Definition in_or_not_in (U : Type) (x : U) (A : Ensemble U) :=
classic (In U A x).
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(**  A and B are sets of elements in the univers U                          *)
Variable U : Type.
Let SU := Ensemble U.
Variable A B : SU.  (* A and B are sets of elements in the univers U *)
Section Bijection.
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(** We now show that if f and g are injections resp from A to B and from  *)
(** B to A, then there is a subset J of A s.t. h, defined to be f on A    *)
(** and the converse of g on A\J is a bijection from A to B               *)
Variable f g : Relation U.  (* f and g are relations *)
Hypothesis f_inj : injection U A B f. (* f and g are injections *)
Hypothesis g_inj : injection U B A g.
Let Imf : Ensemble U -> Ensemble U := Im U f.
Let Img : Ensemble U -> Ensemble U := Im U g.
(** Constructing J s.t. g(B\f(J))=A\J *)
(** (Setminus U A C) denotes the difference A\C         *)
(** (Included U A C) means that A is included in C  *)
Let F (C : SU) := Setminus U A (Img (Setminus U B (Imf C))).
Let D (C : SU) := Included U C (F C).
Let J := Set_Sum U D.
(**  We show that so-built J is the subset we are looking for *)
(** J is Tarski's fix-point of F, a function which is growing *)
(** w.r.t. inclusion                                          *)
(** Lemma: F is growing *)

Lemma F_growing :
forall C C' : SU, Included U C C' -> Included U (F C) (F C').