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.

  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*.

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).

Reflexive closure is included in the reflexive-transitive closure

    Lemma clos_r_clos_rt :
      inclusion (clos_refl R) (clos_refl_trans R).

    Lemma clos_rt_t : forall x y z,
      clos_refl_trans R x y -> clos_trans R y z ->
      clos_trans R x z.

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.

  Lemma clos_trans_transp_permute : forall x y,
    transp _ (clos_trans R) x y <-> clos_trans (transp _ R) x y.

End Properties.