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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 $Id$ i*)
(****************************************************************************)
(* Bruno Barras *)
(****************************************************************************)
Require Relation_Definitions.
Require Relation_Operators.
Section Properties.
Variable A: Set.
Variable R: (relation A).
Local incl : (relation A)->(relation A)->Prop :=
[R1,R2: (relation A)] (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)).
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)).
Red.
NewInduction 1; Auto with sets.
Intros.
Apply rt_trans with y; Auto with sets.
Qed.
Lemma clos_refl_trans_ind_left: (A:Set)(R:A->A->Prop)(M:A)(P:A->Prop)
(P M)
->((P0,N:A)
(clos_refl_trans A R M P0)->(P P0)->(R P0 N)->(P N))
->(a:A)(clos_refl_trans A R M a)->(P a).
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)).
Red.
NewInduction 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)).
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)).
Red.
NewInduction 1; Auto with sets.
Apply rst_trans with y; Auto with sets.
Qed.
End Clos_Refl_Sym_Trans.
End Properties.
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