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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Operators_Properties.v 9597 2007-02-06 19:44:05Z herbelin $ i*)
(****************************************************************************)
(* Bruno Barras *)
(****************************************************************************)
Require Import Relation_Definitions.
Require Import Relation_Operators.
Section Properties.
Variable A : Type.
Variable R : relation A.
Let incl (R1 R2:relation A) : Prop := forall x y:A, R1 x y -> R2 x y.
Section Clos_Refl_Trans.
Lemma clos_rt_is_preorder : preorder A (clos_refl_trans A R).
Proof.
apply Build_preorder.
exact (rt_refl A R).
exact (rt_trans A R).
Qed.
Lemma clos_rt_idempotent :
incl (clos_refl_trans A (clos_refl_trans A R)) (clos_refl_trans A R).
Proof.
red in |- *.
induction 1; auto with sets.
intros.
apply rt_trans with y; auto with sets.
Qed.
Lemma clos_refl_trans_ind_left :
forall (A:Type) (R:A -> A -> Prop) (M:A) (P:A -> Prop),
P M ->
(forall P0 N:A, clos_refl_trans A R M P0 -> P P0 -> R P0 N -> P N) ->
forall a:A, clos_refl_trans A R M a -> P a.
Proof.
intros.
generalize H H0.
clear H H0.
elim H1; intros; auto with sets.
apply H2 with x; auto with sets.
apply H3.
apply H0; auto with sets.
intros.
apply H5 with P0; auto with sets.
apply rt_trans with y; auto with sets.
Qed.
End Clos_Refl_Trans.
Section Clos_Refl_Sym_Trans.
Lemma clos_rt_clos_rst :
inclusion A (clos_refl_trans A R) (clos_refl_sym_trans A R).
Proof.
red in |- *.
induction 1; auto with sets.
apply rst_trans with y; auto with sets.
Qed.
Lemma clos_rst_is_equiv : equivalence A (clos_refl_sym_trans A R).
Proof.
apply Build_equivalence.
exact (rst_refl A R).
exact (rst_trans A R).
exact (rst_sym A R).
Qed.
Lemma clos_rst_idempotent :
incl (clos_refl_sym_trans A (clos_refl_sym_trans A R))
(clos_refl_sym_trans A R).
Proof.
red in |- *.
induction 1; auto with sets.
apply rst_trans with y; auto with sets.
Qed.
End Clos_Refl_Sym_Trans.
End Properties.
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