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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
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(** * Some properties of the operators on relations *)
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(** * Initial version by Bruno Barras *)
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Require Import Relation_Definitions.
Require Import Relation_Operators.
Section Properties.
Arguments clos_refl [A] R x _.
Arguments clos_refl_trans [A] R x _.
Arguments clos_refl_trans_1n [A] R x _.
Arguments clos_refl_trans_n1 [A] R x _.
Arguments clos_refl_sym_trans [A] R _ _.
Arguments clos_refl_sym_trans_1n [A] R x _.
Arguments clos_refl_sym_trans_n1 [A] R x _.
Arguments clos_trans [A] R x _.
Arguments clos_trans_1n [A] R x _.
Arguments clos_trans_n1 [A] R x _.
Arguments inclusion [A] R1 R2.
Arguments preorder [A] R.
Variable A : Type.
Variable R : relation A.
Section Clos_Refl_Trans.
Local Notation "R *" := (clos_refl_trans R)
(at level 8, no associativity, format "R *").
(** Correctness of the reflexive-transitive closure operator *)
Lemma clos_rt_is_preorder : preorder R*.
Proof.
apply Build_preorder.
- exact (rt_refl A R).
- exact (rt_trans A R).
Qed.
(** Idempotency of the reflexive-transitive closure operator *)
Lemma clos_rt_idempotent : inclusion (R*)* R*.
Proof.
red.
induction 1; auto with sets.
intros.
apply rt_trans with y; auto with sets.
Qed.
End Clos_Refl_Trans.
Section Clos_Refl_Sym_Trans.
(** Reflexive-transitive closure is included in the
reflexive-symmetric-transitive closure *)
Lemma clos_rt_clos_rst :
inclusion (clos_refl_trans R) (clos_refl_sym_trans R).
Proof.
red.
induction 1; auto with sets.
apply rst_trans with y; auto with sets.
Qed.
(** Reflexive closure is included in the
reflexive-transitive closure *)
Lemma clos_r_clos_rt :
inclusion (clos_refl R) (clos_refl_trans R).
Proof.
induction 1 as [? ?| ].
- constructor; auto.
- constructor 2.
Qed.
Lemma clos_rt_t : forall x y z,
clos_refl_trans R x y -> clos_trans R y z ->
clos_trans R x z.
Proof.
induction 1 as [b d H1|b|a b d H1 H2 IH1 IH2]; auto.
intro H. apply t_trans with (y:=d); auto.
constructor. auto.
Qed.
(** Correctness of the reflexive-symmetric-transitive closure *)
Lemma clos_rst_is_equiv : equivalence A (clos_refl_sym_trans R).
Proof.
apply Build_equivalence.
- exact (rst_refl A R).
- exact (rst_trans A R).
- exact (rst_sym A R).
Qed.
(** Idempotency of the reflexive-symmetric-transitive closure operator *)
Lemma clos_rst_idempotent :
inclusion (clos_refl_sym_trans (clos_refl_sym_trans R))
(clos_refl_sym_trans R).
Proof.
red.
induction 1; auto with sets.
apply rst_trans with y; auto with sets.
Qed.
End Clos_Refl_Sym_Trans.
Section Equivalences.
(** *** Equivalences between the different definition of the reflexive,
symmetric, transitive closures *)
(** *** Contributed by P. Castéran *)
(** Direct transitive closure vs left-step extension *)
Lemma clos_t1n_trans : forall x y, clos_trans_1n R x y -> clos_trans R x y.
Proof.
induction 1.
- left; assumption.
- right with y; auto.
left; auto.
Qed.
Lemma clos_trans_t1n : forall x y, clos_trans R x y -> clos_trans_1n R x y.
Proof.
induction 1.
- left; assumption.
- generalize IHclos_trans2; clear IHclos_trans2; induction IHclos_trans1.
+ right with y; auto.
+ right with y; auto.
eapply IHIHclos_trans1; auto.
apply clos_t1n_trans; auto.
Qed.
Lemma clos_trans_t1n_iff : forall x y,
clos_trans R x y <-> clos_trans_1n R x y.
Proof.
split.
- apply clos_trans_t1n.
- apply clos_t1n_trans.
Qed.
(** Direct transitive closure vs right-step extension *)
Lemma clos_tn1_trans : forall x y, clos_trans_n1 R x y -> clos_trans R x y.
Proof.
induction 1.
- left; assumption.
- right with y; auto.
left; assumption.
Qed.
Lemma clos_trans_tn1 : forall x y, clos_trans R x y -> clos_trans_n1 R x y.
Proof.
induction 1.
- left; assumption.
- elim IHclos_trans2.
+ intro y0; right with y.
* auto.
* auto.
+ intros.
right with y0; auto.
Qed.
Lemma clos_trans_tn1_iff : forall x y,
clos_trans R x y <-> clos_trans_n1 R x y.
Proof.
split.
- apply clos_trans_tn1.
- apply clos_tn1_trans.
Qed.
(** Direct reflexive-transitive closure is equivalent to
transitivity by left-step extension *)
Lemma clos_rt1n_step : forall x y, R x y -> clos_refl_trans_1n R x y.
Proof.
intros x y H.
right with y;[assumption|left].
Qed.
Lemma clos_rtn1_step : forall x y, R x y -> clos_refl_trans_n1 R x y.
Proof.
intros x y H.
right with x;[assumption|left].
Qed.
Lemma clos_rt1n_rt : forall x y,
clos_refl_trans_1n R x y -> clos_refl_trans R x y.
Proof.
induction 1.
- constructor 2.
- constructor 3 with y; auto.
constructor 1; auto.
Qed.
Lemma clos_rt_rt1n : forall x y,
clos_refl_trans R x y -> clos_refl_trans_1n R x y.
Proof.
induction 1.
- apply clos_rt1n_step; assumption.
- left.
- generalize IHclos_refl_trans2; clear IHclos_refl_trans2;
induction IHclos_refl_trans1; auto.
right with y; auto.
eapply IHIHclos_refl_trans1; auto.
apply clos_rt1n_rt; auto.
Qed.
Lemma clos_rt_rt1n_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_1n R x y.
Proof.
split.
- apply clos_rt_rt1n.
- apply clos_rt1n_rt.
Qed.
(** Direct reflexive-transitive closure is equivalent to
transitivity by right-step extension *)
Lemma clos_rtn1_rt : forall x y,
clos_refl_trans_n1 R x y -> clos_refl_trans R x y.
Proof.
induction 1.
- constructor 2.
- constructor 3 with y; auto.
constructor 1; assumption.
Qed.
Lemma clos_rt_rtn1 : forall x y,
clos_refl_trans R x y -> clos_refl_trans_n1 R x y.
Proof.
induction 1.
- apply clos_rtn1_step; auto.
- left.
- elim IHclos_refl_trans2; auto.
intros.
right with y0; auto.
Qed.
Lemma clos_rt_rtn1_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_n1 R x y.
Proof.
split.
- apply clos_rt_rtn1.
- apply clos_rtn1_rt.
Qed.
(** Induction on the left transitive step *)
Lemma clos_refl_trans_ind_left :
forall (x:A) (P:A -> Prop), P x ->
(forall y z:A, clos_refl_trans R x y -> P y -> R y z -> P z) ->
forall z:A, clos_refl_trans R x z -> P z.
Proof.
intros.
revert H H0.
induction H1; intros; auto with sets.
- apply H1 with x; auto with sets.
- apply IHclos_refl_trans2.
+ apply IHclos_refl_trans1; auto with sets.
+ intros.
apply H0 with y0; auto with sets.
apply rt_trans with y; auto with sets.
Qed.
(** Induction on the right transitive step *)
Lemma rt1n_ind_right : forall (P : A -> Prop) (z:A),
P z ->
(forall x y, R x y -> clos_refl_trans_1n R y z -> P y -> P x) ->
forall x, clos_refl_trans_1n R x z -> P x.
induction 3; auto.
apply H0 with y; auto.
Qed.
Lemma clos_refl_trans_ind_right : forall (P : A -> Prop) (z:A),
P z ->
(forall x y, R x y -> P y -> clos_refl_trans R y z -> P x) ->
forall x, clos_refl_trans R x z -> P x.
intros P z Hz IH x Hxz.
apply clos_rt_rt1n_iff in Hxz.
elim Hxz using rt1n_ind_right; auto.
clear x Hxz.
intros x y Hxy Hyz Hy.
apply clos_rt_rt1n_iff in Hyz.
eauto.
Qed.
(** Direct reflexive-symmetric-transitive closure is equivalent to
transitivity by symmetric left-step extension *)
Lemma clos_rst1n_rst : forall x y,
clos_refl_sym_trans_1n R x y -> clos_refl_sym_trans R x y.
Proof.
induction 1.
- constructor 2.
- constructor 4 with y; auto.
case H;[constructor 1|constructor 3; constructor 1]; auto.
Qed.
Lemma clos_rst1n_trans : forall x y z, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y z -> clos_refl_sym_trans_1n R x z.
induction 1.
- auto.
- intros; right with y; eauto.
Qed.
Lemma clos_rst1n_sym : forall x y, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y x.
Proof.
intros x y H; elim H.
- constructor 1.
- intros x0 y0 z D H0 H1; apply clos_rst1n_trans with y0; auto.
right with x0.
+ tauto.
+ left.
Qed.
Lemma clos_rst_rst1n : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_1n R x y.
induction 1.
- constructor 2 with y; auto.
constructor 1.
- constructor 1.
- apply clos_rst1n_sym; auto.
- eapply clos_rst1n_trans; eauto.
Qed.
Lemma clos_rst_rst1n_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_1n R x y.
Proof.
split.
- apply clos_rst_rst1n.
- apply clos_rst1n_rst.
Qed.
(** Direct reflexive-symmetric-transitive closure is equivalent to
transitivity by symmetric right-step extension *)
Lemma clos_rstn1_rst : forall x y,
clos_refl_sym_trans_n1 R x y -> clos_refl_sym_trans R x y.
Proof.
induction 1.
- constructor 2.
- constructor 4 with y; auto.
case H;[constructor 1|constructor 3; constructor 1]; auto.
Qed.
Lemma clos_rstn1_trans : forall x y z, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y z -> clos_refl_sym_trans_n1 R x z.
Proof.
intros x y z H1 H2.
induction H2.
- auto.
- intros.
right with y0; eauto.
Qed.
Lemma clos_rstn1_sym : forall x y, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y x.
Proof.
intros x y H; elim H.
- constructor 1.
- intros y0 z D H0 H1. apply clos_rstn1_trans with y0; auto.
right with z.
+ tauto.
+ left.
Qed.
Lemma clos_rst_rstn1 : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_n1 R x y.
Proof.
induction 1.
- constructor 2 with x; auto.
constructor 1.
- constructor 1.
- apply clos_rstn1_sym; auto.
- eapply clos_rstn1_trans; eauto.
Qed.
Lemma clos_rst_rstn1_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_n1 R x y.
Proof.
split.
- apply clos_rst_rstn1.
- apply clos_rstn1_rst.
Qed.
End Equivalences.
Lemma clos_trans_transp_permute : forall x y,
transp _ (clos_trans R) x y <-> clos_trans (transp _ R) x y.
Proof.
split; induction 1;
(apply t_step; assumption) || eapply t_trans; eassumption.
Qed.
End Properties.
(* begin hide *)
(* Compatibility *)
Notation trans_tn1 := clos_trans_tn1 (only parsing).
Notation tn1_trans := clos_tn1_trans (only parsing).
Notation tn1_trans_equiv := clos_trans_tn1_iff (only parsing).
Notation trans_t1n := clos_trans_t1n (only parsing).
Notation t1n_trans := clos_t1n_trans (only parsing).
Notation t1n_trans_equiv := clos_trans_t1n_iff (only parsing).
Notation R_rtn1 := clos_rtn1_step (only parsing).
Notation trans_rt1n := clos_rt_rt1n (only parsing).
Notation rt1n_trans := clos_rt1n_rt (only parsing).
Notation rt1n_trans_equiv := clos_rt_rt1n_iff (only parsing).
Notation R_rt1n := clos_rt1n_step (only parsing).
Notation trans_rtn1 := clos_rt_rtn1 (only parsing).
Notation rtn1_trans := clos_rtn1_rt (only parsing).
Notation rtn1_trans_equiv := clos_rt_rtn1_iff (only parsing).
Notation rts1n_rts := clos_rst1n_rst (only parsing).
Notation rts_1n_trans := clos_rst1n_trans (only parsing).
Notation rts1n_sym := clos_rst1n_sym (only parsing).
Notation rts_rts1n := clos_rst_rst1n (only parsing).
Notation rts_rts1n_equiv := clos_rst_rst1n_iff (only parsing).
Notation rtsn1_rts := clos_rstn1_rst (only parsing).
Notation rtsn1_trans := clos_rstn1_trans (only parsing).
Notation rtsn1_sym := clos_rstn1_sym (only parsing).
Notation rts_rtsn1 := clos_rst_rstn1 (only parsing).
Notation rts_rtsn1_equiv := clos_rst_rstn1_iff (only parsing).
(* end hide *)
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