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.
Require Import Setoid.

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.

Correctness of the reflexive-transitive closure operator

    Lemma clos_rt_is_preorder : preorder A (clos_refl_trans A R).

Idempotency of the reflexive-transitive closure operator

    Lemma clos_rt_idempotent :
      incl (clos_refl_trans A (clos_refl_trans A R)) (clos_refl_trans A 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 A (clos_refl_trans A R) (clos_refl_sym_trans A R).

Correctness of the reflexive-symmetric-transitive closure
Idempotency of the reflexive-symmetric-transitive closure operator

    Lemma clos_rst_idempotent :
      incl (clos_refl_sym_trans A (clos_refl_sym_trans A R))
      (clos_refl_sym_trans A R).

  End Clos_Refl_Sym_Trans.

  Section Equivalences.

Equivalences between the different definition of the reflexive,

symmetric, transitive closures

Contributed by P. Casteran


Direct transitive closure vs left-step extension

    Lemma t1n_trans : forall x y, clos_trans_1n A R x y -> clos_trans A R x y.

    Lemma trans_t1n : forall x y, clos_trans A R x y -> clos_trans_1n A R x y.

    Lemma t1n_trans_equiv : forall x y,
        clos_trans A R x y <-> clos_trans_1n A R x y.

Direct transitive closure vs right-step extension

    Lemma tn1_trans : forall x y, clos_trans_n1 A R x y -> clos_trans A R x y.

    Lemma trans_tn1 : forall x y, clos_trans A R x y -> clos_trans_n1 A R x y.

    Lemma tn1_trans_equiv : forall x y,
        clos_trans A R x y <-> clos_trans_n1 A R x y.

Direct reflexive-transitive closure is equivalent to transitivity by left-step extension

    Lemma R_rt1n : forall x y, R x y -> clos_refl_trans_1n A R x y.

    Lemma R_rtn1 : forall x y, R x y -> clos_refl_trans_n1 A R x y.

    Lemma rt1n_trans : forall x y,
        clos_refl_trans_1n A R x y -> clos_refl_trans A R x y.

    Lemma trans_rt1n : forall x y,
        clos_refl_trans A R x y -> clos_refl_trans_1n A R x y.

    Lemma rt1n_trans_equiv : forall x y,
      clos_refl_trans A R x y <-> clos_refl_trans_1n A R x y.

Direct reflexive-transitive closure is equivalent to transitivity by right-step extension

    Lemma rtn1_trans : forall x y,
        clos_refl_trans_n1 A R x y -> clos_refl_trans A R x y.

    Lemma trans_rtn1 : forall x y,
        clos_refl_trans A R x y -> clos_refl_trans_n1 A R x y.

    Lemma rtn1_trans_equiv : forall x y,
        clos_refl_trans A R x y <-> clos_refl_trans_n1 A 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 A R x y -> P y -> R y z -> P z) ->
        forall z:A, clos_refl_trans A 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 A R y z -> P y -> P x) ->
      forall x, clos_refl_trans_1n A 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 A R y z -> P x) ->
      forall x, clos_refl_trans A R x z -> P x.

Direct reflexive-symmetric-transitive closure is equivalent to transitivity by symmetric left-step extension

    Lemma rts1n_rts : forall x y,
      clos_refl_sym_trans_1n A R x y -> clos_refl_sym_trans A R x y.

    Lemma rts_1n_trans : forall x y, clos_refl_sym_trans_1n A R x y ->
      forall z, clos_refl_sym_trans_1n A R y z ->
        clos_refl_sym_trans_1n A R x z.

    Lemma rts1n_sym : forall x y, clos_refl_sym_trans_1n A R x y ->
      clos_refl_sym_trans_1n A R y x.

    Lemma rts_rts1n : forall x y,
      clos_refl_sym_trans A R x y -> clos_refl_sym_trans_1n A R x y.

    Lemma rts_rts1n_equiv : forall x y,
      clos_refl_sym_trans A R x y <-> clos_refl_sym_trans_1n A R x y.

Direct reflexive-symmetric-transitive closure is equivalent to transitivity by symmetric right-step extension

    Lemma rtsn1_rts : forall x y,
      clos_refl_sym_trans_n1 A R x y -> clos_refl_sym_trans A R x y.

    Lemma rtsn1_trans : forall y z, clos_refl_sym_trans_n1 A R y z->
      forall x, clos_refl_sym_trans_n1 A R x y ->
        clos_refl_sym_trans_n1 A R x z.

    Lemma rtsn1_sym : forall x y, clos_refl_sym_trans_n1 A R x y ->
      clos_refl_sym_trans_n1 A R y x.


    Lemma rts_rtsn1 : forall x y,
      clos_refl_sym_trans A R x y -> clos_refl_sym_trans_n1 A R x y.

    Lemma rts_rtsn1_equiv : forall x y,
      clos_refl_sym_trans A R x y <-> clos_refl_sym_trans_n1 A R x y.

  End Equivalences.

End Properties.