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Some cleanup for the extensionality proof

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Dan Frumin 2017-06-19 12:24:57 +02:00
parent dce70f517f
commit 490980db0f
1 changed files with 37 additions and 39 deletions
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76
FiniteSets/properties.v
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@ -437,6 +437,7 @@ Proof.
reflexivity. reflexivity.
Defined. Defined.
(* Proof of the extensionality axiom *)
(* Properties about subset relation. *) (* Properties about subset relation. *)
Lemma subset_union `{Funext} (X Y : FSet A) : Lemma subset_union `{Funext} (X Y : FSet A) :
subset X Y = true -> U X Y = Y. subset X Y = true -> U X Y = Y.
@ -472,17 +473,6 @@ hinduction X; try (intros; apply path_forall; intro; apply set_path2).
* contradiction (false_ne_true). * contradiction (false_ne_true).
Defined. Defined.
Lemma eq1 (X Y : FSet A) : X = Y <~> (U Y X = X) * (U X Y = Y).
Proof.
eapply equiv_iff_hprop_uncurried.
split.
- intro H. split.
apply (comm Y X @ ap (U X) H^ @ union_idem X).
apply (ap (U X) H^ @ union_idem X @ H).
- intros [H1 H2]. etransitivity. apply H1^.
apply (comm Y X @ H2).
Defined.
Lemma subset_union_l `{Funext} X : Lemma subset_union_l `{Funext} X :
forall Y, subset X (U X Y) = true. forall Y, subset X (U X Y) = true.
Proof. Proof.
@ -493,13 +483,10 @@ hinduction X;
* reflexivity. * reflexivity.
* by contradiction n. * by contradiction n.
- intros X1 X2 HX1 HX2 Y. - intros X1 X2 HX1 HX2 Y.
enough (subset X1 (U (U X1 X2) Y) = true). refine (ap (fun z => (X1 ⊆ z && X2 ⊆ (X1 ∪ X2) ∪ Y))%Bool (assoc X1 X2 Y)^ @ _).
enough (subset X2 (U (U X1 X2) Y) = true). refine (ap (fun z => (X1 ⊆ _ && X2 ⊆ z ∪ Y))%Bool (comm _ _) @ _).
rewrite X. rewrite X0. reflexivity. refine (ap (fun z => (X1 ⊆ _ && X2 ⊆ z))%Bool (assoc _ _ _)^ @ _).
{ rewrite (comm X1 X2). rewrite HX1. simpl. apply HX2.
rewrite <- (assoc X2 X1 Y).
apply (HX2 (U X1 Y)). }
{ rewrite <- (assoc X1 X2 Y). apply (HX1 (U X2 Y)). }
Defined. Defined.
Lemma subset_union_equiv `{Funext} Lemma subset_union_equiv `{Funext}
@ -514,11 +501,22 @@ Proof.
apply subset_union_l. apply subset_union_l.
Defined. Defined.
Lemma eq_subset' (X Y : FSet A) : X = Y <~> (U Y X = X) * (U X Y = Y).
Proof.
eapply equiv_iff_hprop_uncurried.
split.
- intro H. split.
apply (comm Y X @ ap (U X) H^ @ union_idem X).
apply (ap (U X) H^ @ union_idem X @ H).
- intros [H1 H2]. etransitivity. apply H1^.
apply (comm Y X @ H2).
Defined.
Lemma eq_subset `{Funext} (X Y : FSet A) : Lemma eq_subset `{Funext} (X Y : FSet A) :
X = Y <~> ((subset Y X = true) * (subset X Y = true)). X = Y <~> ((subset Y X = true) * (subset X Y = true)).
Proof. Proof.
transitivity ((U Y X = X) * (U X Y = Y)). transitivity ((U Y X = X) * (U X Y = Y)).
apply eq1. apply eq_subset'.
symmetry. symmetry.
eapply equiv_functor_prod'; apply subset_union_equiv. eapply equiv_functor_prod'; apply subset_union_equiv.
Defined. Defined.
@ -557,26 +555,26 @@ Proof.
destruct (dec (a = b)). destruct (dec (a = b)).
+ intros ; rewrite p ; apply H. + intros ; rewrite p ; apply H.
+ intros X ; contradiction (false_ne_true X). + intros X ; contradiction (false_ne_true X).
* intros X1 X2. * intros X1 X2.
intros IH1 IH2 H1 a H2. intros IH1 IH2 H1 a H2.
destruct (subset X1 Y) ; destruct (subset X2 Y); destruct (subset X1 Y) ; destruct (subset X2 Y);
cbv in H1; try by contradiction false_ne_true. cbv in H1; try by contradiction false_ne_true.
specialize (IH1 idpath a). specialize (IH2 idpath a). specialize (IH1 idpath a). specialize (IH2 idpath a).
destruct (isIn a X1); destruct (isIn a X2); destruct (isIn a X1); destruct (isIn a X2);
cbv in H2; try by contradiction false_ne_true. cbv in H2; try by contradiction false_ne_true.
by apply IH1. by apply IH1.
by apply IH1. by apply IH1.
by apply IH2. by apply IH2.
* repeat (intro; intros; apply path_forall). * repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2. intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall). * repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2. intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall). * repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2. intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall). * repeat (intro; intros; apply path_forall).
intros; intro; intros; apply set_path2. intros; intro; intros; apply set_path2.
* repeat (intro; intros; apply path_forall); * repeat (intro; intros; apply path_forall);
intros; intro; intros; apply set_path2. intros; intro; intros; apply set_path2.
Defined. Defined.
Theorem fset_ext `{Funext} (X Y : FSet A) : Theorem fset_ext `{Funext} (X Y : FSet A) :