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Added idempotency of the intersection

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Niels
2017-05-23 21:31:45 +02:00
parent 13737556c6
commit 6bb5e8b690
2 changed files with 221 additions and 75 deletions
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47
FiniteSets/properties.v
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@@ -265,7 +265,50 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
cbn.
rewrite comprehension_or.
reflexivity.
*)
*)
End properties.
Theorem comprehension_subset : forall ϕ (X : FSet A),
U (comprehension ϕ X) X = X.
Proof.
intros ϕ.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- apply nl.
- intro a.
destruct (ϕ a).
* apply union_idem.
* apply nl.
- intros X Y P Q.
rewrite assoc.
rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
rewrite (comm (comprehension ϕ Y) X).
rewrite assoc.
rewrite P.
rewrite <- assoc.
rewrite Q.
reflexivity.
Defined.
Theorem intersection_idem : forall (X : FSet A), intersection X X = X.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- reflexivity.
- intro a.
unfold operations.deceq.
destruct (dec (a = a)).
* reflexivity.
* contradiction (n idpath).
- intros X Y IHX IHY.
cbn in *.
rewrite comprehension_or.
rewrite comprehension_or.
unfold intersection in *.
rewrite IHX.
rewrite IHY.
rewrite comprehension_subset.
rewrite (comm X).
rewrite comprehension_subset.
reflexivity.
Defined.
End properties.