Library Stdlib.Wellfounded.Transitive_Closure

Require Import Relation_Definitions.
Require Import Relation_Operators.

Section Wf_Transitive_Closure.
  Variable A : Type.
  Variable R : relation A.

  Notation trans_clos := (clos_trans A R).

  Lemma incl_clos_trans : inclusion A R trans_clos.

  Lemma Acc_clos_trans : forall x:A, Acc R x -> Acc trans_clos x.

  #[local]
  Hint Resolve Acc_clos_trans : core.

  Lemma Acc_inv_trans : forall x y:A, trans_clos y x -> Acc R x -> Acc R y.

  Theorem wf_clos_trans : well_founded R -> well_founded trans_clos.

End Wf_Transitive_Closure.