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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(****************************************************************************)
(* *)
(* Naive Set Theory in Coq *)
(* *)
(* INRIA INRIA *)
(* Rocquencourt Sophia-Antipolis *)
(* *)
(* Coq V6.1 *)
(* *)
(* Gilles Kahn *)
(* Gerard Huet *)
(* *)
(* *)
(* *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *)
(* to the Newton Institute for providing an exceptional work environment *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *)
(****************************************************************************)
Require Export Relations_1.
Definition Complement (U:Type) (R:Relation U) : Relation U :=
fun x y:U => ~ R x y.
Theorem Rsym_imp_notRsym :
forall (U:Type) (R:Relation U),
Symmetric U R -> Symmetric U (Complement U R).
Proof.
unfold Symmetric, Complement.
intros U R H' x y H'0; red; intro H'1; apply H'0; auto with sets.
Qed.
Theorem Equiv_from_preorder :
forall (U:Type) (R:Relation U),
Preorder U R -> Equivalence U (fun x y:U => R x y /\ R y x).
Proof.
intros U R H'; elim H'; intros H'0 H'1.
apply Definition_of_equivalence.
red in H'0; auto 10 with sets.
2: red; intros x y h; elim h; intros H'3 H'4; auto 10 with sets.
red in H'1; red; auto 10 with sets.
intros x y z h; elim h; intros H'3 H'4; clear h.
intro h; elim h; intros H'5 H'6; clear h.
split; apply H'1 with y; auto 10 with sets.
Qed.
Hint Resolve Equiv_from_preorder.
Theorem Equiv_from_order :
forall (U:Type) (R:Relation U),
Order U R -> Equivalence U (fun x y:U => R x y /\ R y x).
Proof.
intros U R H'; elim H'; auto 10 with sets.
Qed.
Hint Resolve Equiv_from_order.
Theorem contains_is_preorder :
forall U:Type, Preorder (Relation U) (contains U).
Proof.
auto 10 with sets.
Qed.
Hint Resolve contains_is_preorder.
Theorem same_relation_is_equivalence :
forall U:Type, Equivalence (Relation U) (same_relation U).
Proof.
unfold same_relation at 1; auto 10 with sets.
Qed.
Hint Resolve same_relation_is_equivalence.
Theorem cong_reflexive_same_relation :
forall (U:Type) (R R':Relation U),
same_relation U R R' -> Reflexive U R -> Reflexive U R'.
Proof.
unfold same_relation; intuition.
Qed.
Theorem cong_symmetric_same_relation :
forall (U:Type) (R R':Relation U),
same_relation U R R' -> Symmetric U R -> Symmetric U R'.
Proof.
compute; intros; elim H; intros; clear H;
apply (H3 y x (H0 x y (H2 x y H1))).
(*Intuition.*)
Qed.
Theorem cong_antisymmetric_same_relation :
forall (U:Type) (R R':Relation U),
same_relation U R R' -> Antisymmetric U R -> Antisymmetric U R'.
Proof.
compute; intros; elim H; intros; clear H;
apply (H0 x y (H3 x y H1) (H3 y x H2)).
(*Intuition.*)
Qed.
Theorem cong_transitive_same_relation :
forall (U:Type) (R R':Relation U),
same_relation U R R' -> Transitive U R -> Transitive U R'.
Proof.
intros U R R' H' H'0; red.
elim H'.
intros H'1 H'2 x y z H'3 H'4; apply H'2.
apply H'0 with y; auto with sets.
Qed.
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