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A proof of subset_isIn

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Dan Frumin 2017-05-25 18:59:18 +02:00
parent 294f818b07
commit 06dc8a0acd
1 changed files with 55 additions and 4 deletions
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59
FiniteSets/properties.v
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@ -524,10 +524,61 @@ Proof.
eapply equiv_functor_prod'; apply subset_union_equiv.
Defined.
Lemma subset_isIn (X Y : FSet A) :
Lemma subset_isIn `{FE : Funext} (X Y : FSet A) :
(forall (a : A), isIn a X = true -> isIn a Y = true)
<-> (subset X Y = true).
Admitted.
Proof.
split.
- hinduction X ; try (intros ; apply path_forall ; intro ; apply set_path2).
* intros ; reflexivity.
* intros a H.
apply H.
destruct (dec (a = a)).
+ reflexivity.
+ contradiction (n idpath).
* intros X1 X2 H1 H2 H.
enough (subset X1 Y = true).
rewrite X.
enough (subset X2 Y = true).
rewrite X0.
reflexivity.
+ apply H2.
intros a Ha.
apply H.
rewrite Ha.
destruct (isIn a X1) ; reflexivity.
+ apply H1.
intros a Ha.
apply H.
rewrite Ha.
reflexivity.
- hinduction X .
* intros. contradiction (false_ne_true X0).
* intros b H a.
destruct (dec (a = b)).
+ intros ; rewrite p ; apply H.
+ intros X ; contradiction (false_ne_true X).
* intros X1 X2.
intros IH1 IH2 H1 a H2.
destruct (subset X1 Y) ; destruct (subset X2 Y);
cbv in H1; try by contradiction false_ne_true.
specialize (IH1 idpath a). specialize (IH2 idpath a).
destruct (isIn a X1); destruct (isIn a X2);
cbv in H2; try by contradiction false_ne_true.
by apply IH1.
by apply IH1.
by apply IH2.
* repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall);
intros; intro; intros; apply set_path2.
Defined.
Theorem fset_ext `{Funext} (X Y : FSet A) :
X = Y <~> (forall (a : A), isIn a X = isIn a Y).
@ -536,8 +587,8 @@ Proof.
transitivity
((forall a, isIn a Y = true -> isIn a X = true)
*(forall a, isIn a X = true -> isIn a Y = true)).
- admit.
- admit.
- eapply equiv_functor_prod'. admit. admit.
- eapply equiv_functor_prod'.
Admitted.
End properties.