Library Coq.Wellfounded.Transitive_Closure

Author: Bruno Barras




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.

    red in |- *; auto with sets.

  Qed.




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

    induction 1 as [x0 _ H1].

    apply Acc_intro.

    intros y H2.

    induction H2; auto with sets.

    apply Acc_inv with y; auto with sets.

  Qed.




  Hint Resolve Acc_clos_trans.




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

  Proof.

    induction 1 as [| x y]; auto with sets.

    intro; apply Acc_inv with y; assumption.

  Qed.




  Theorem wf_clos_trans : well_founded R -> well_founded trans_clos.

  Proof.

    unfold well_founded in |- *; auto with sets.

  Qed.




End Wf_Transitive_Closure.