# Library Coq.Relations.Operators_Properties

# Some properties of the operators on relations

# Initial version by Bruno Barras

Require Import Relation_Definitions.
Require Import Relation_Operators.
Section Properties.
Implicit Arguments clos_refl_trans [A].
Implicit Arguments clos_refl_trans_1n [A].
Implicit Arguments clos_refl_trans_n1 [A].
Implicit Arguments clos_refl_sym_trans [A].
Implicit Arguments clos_refl_sym_trans_1n [A].
Implicit Arguments clos_refl_sym_trans_n1 [A].
Implicit Arguments clos_trans [A].
Implicit Arguments clos_trans_1n [A].
Implicit Arguments clos_trans_n1 [A].
Implicit Arguments inclusion [A].
Implicit Arguments preorder [A].
Variable A : Type.
Variable R : relation A.
Section Clos_Refl_Trans.
Local Notation "R *" := (clos_refl_trans R) (at level 8, left associativity).
Correctness of the reflexive-transitive closure operator
Lemma clos_rt_is_preorder : preorder R *.
Idempotency of the reflexive-transitive closure operator
Lemma clos_rt_idempotent : inclusion ( R *)* R *.
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).
Correctness of the reflexive-symmetric-transitive closure
Lemma clos_rst_is_equiv : equivalence A (clos_refl_sym_trans R).
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).
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.
Lemma clos_trans_t1n : forall x y, clos_trans R x y -> clos_trans_1n R x y.
Lemma clos_trans_t1n_iff : forall x y,
clos_trans R x y <-> clos_trans_1n R x y.
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.
Lemma clos_trans_tn1 : forall x y, clos_trans R x y -> clos_trans_n1 R x y.
Lemma clos_trans_tn1_iff : forall x y,
clos_trans R x y <-> clos_trans_n1 R x y.
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.
Lemma clos_rtn1_step : forall x y, R x y -> clos_refl_trans_n1 R x y.
Lemma clos_rt1n_rt : forall x y,
clos_refl_trans_1n R x y -> clos_refl_trans R x y.
Lemma clos_rt_rt1n : forall x y,
clos_refl_trans R x y -> clos_refl_trans_1n R x y.
Lemma clos_rt_rt1n_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_1n R x y.
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.
Lemma clos_rt_rtn1 : forall x y,
clos_refl_trans R x y -> clos_refl_trans_n1 R x y.
Lemma clos_rt_rtn1_iff : forall x y,
clos_refl_trans R x y <-> clos_refl_trans_n1 R x y.
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.
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.
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.
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.
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.
Lemma clos_rst1n_sym : forall x y, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y x.
Lemma clos_rst_rst1n : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_1n R x y.
Lemma clos_rst_rst1n_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_1n R x y.
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.
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.
Lemma clos_rstn1_sym : forall x y, clos_refl_sym_trans_n1 R x y ->
clos_refl_sym_trans_n1 R y x.
Lemma clos_rst_rstn1 : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_n1 R x y.
Lemma clos_rst_rstn1_iff : forall x y,
clos_refl_sym_trans R x y <-> clos_refl_sym_trans_n1 R x y.
End Equivalences.
End Properties.