Inclusion.v

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Author: Bruno Barras

Require Import Relation_Definitions.

Section WfInclusion.
  Variable A : Type.
  Variables R1 R2 : A -> A -> Prop.

  Lemma Acc_incl : inclusion A R1 R2 -> forall z:A, Acc R2 z -> Acc R1 z.A:Type
R1, R2:A -> A -> Prop
inclusion A R1 R2 ->
forall z : A, Acc R2 z -> Acc R1 z
  Proof.A:Type
R1, R2:A -> A -> Prop
inclusion A R1 R2 ->
forall z : A, Acc R2 z -> Acc R1 z
    induction 2.A:Type
R1, R2:A -> A -> Prop
H:inclusion A R1 R2
x:A
H0:forall y : A, R2 y x -> Acc R2 y
H1:forall y : A, R2 y x -> Acc R1 y
Acc R1 x
    apply Acc_intro; auto with sets.
  Qed.

  Hint Resolve Acc_incl : core.

  Theorem wf_incl : inclusion A R1 R2 -> well_founded R2 -> well_founded R1.A:Type
R1, R2:A -> A -> Prop
inclusion A R1 R2 ->
well_founded R2 -> well_founded R1
  Proof.A:Type
R1, R2:A -> A -> Prop
inclusion A R1 R2 ->
well_founded R2 -> well_founded R1
    unfold well_founded; auto with sets.
  Qed.

End WfInclusion.