Library Coq.Wellfounded.Inverse_Image

Author: Bruno Barras




Section Inverse_Image.




  Variables A B : Set.

  Variable R : B -> B -> Prop.

  Variable f : A -> B.




  Let Rof (x y:A) : Prop := R (f x) (f y).




  Remark Acc_lemma : forall y:B, Acc R y -> forall x:A, y = f x -> Acc Rof x.

  Proof.

    induction 1 as [y _ IHAcc]; intros x H.

    apply Acc_intro; intros y0 H1.

    apply (IHAcc (f y0)); try trivial.

    rewrite H; trivial.

  Qed.




  Lemma Acc_inverse_image : forall x:A, Acc R (f x) -> Acc Rof x.

  Proof.

    intros; apply (Acc_lemma (f x)); trivial.

  Qed.




  Theorem wf_inverse_image : well_founded R -> well_founded Rof.

  Proof.

    red in |- *; intros; apply Acc_inverse_image; auto.

  Qed.




  Variable F : A -> B -> Prop.

  Let RoF (x y:A) : Prop :=

    exists2 b : B, F x b & (forall c:B, F y c -> R b c).




  Lemma Acc_inverse_rel : forall b:B, Acc R b -> forall x:A, F x b -> Acc RoF x.

  Proof.

    induction 1 as [x _ IHAcc]; intros x0 H2.

    constructor; intros y H3.

    destruct H3.

    apply (IHAcc x1); auto.

  Qed.







  Theorem wf_inverse_rel : well_founded R -> well_founded RoF.

  Proof.

    red in |- *; constructor; intros.

    case H0; intros.

    apply (Acc_inverse_rel x); auto.

  Qed.




End Inverse_Image.