Skip to content

Wellfounded/Inclusion.v add lemma for when the inclusion is partial #238

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
SkySkimmer wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Wellfounded/Inclusion.v add lemma for when the inclusion is partial #238

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
22 changes: 19 additions & 3 deletions theories/Wellfounded/Inclusion.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -10,24 +10,40 @@

(** Author: Bruno Barras *)

From Stdlib Require Import Relation_Definitions.
From Stdlib Require Import Relation_Definitions Relation_Operators.

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

Lemma Acc_partial_incl : forall z : A,
(forall x y, clos_refl_trans _ R1 y z -> R1 x y -> R2 x y) ->
Acc R2 z -> Acc R1 z.
Proof.
intros z H.
induction 1 as [z Hz IH].
apply Acc_intro.
intros y Hy.
apply IH.
- apply H,Hy.
auto with sets.
- intros x z' Hz' HR.
apply H, HR.
apply rt_trans with y;auto with sets.
Qed.

Lemma Acc_incl : inclusion A R1 R2 -> forall z:A, Acc R2 z -> Acc R1 z.
Proof.
induction 2.
apply Acc_intro; auto with sets.
apply Acc_intro; auto.
Qed.

#[local]
Hint Resolve Acc_incl : core.

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

End WfInclusion.