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Niels
2017-08-07 23:27:53 +02:00
parent 30004e1c8b
commit 76fe6faff2
1 changed files with 18 additions and 29 deletions
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47
FiniteSets/fsets/properties.v
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@@ -139,35 +139,29 @@ Section properties.
Context {B : Type}.
Lemma isIn_singleproduct : forall (a : A) (b : B) (c : A) (Y : FSet B),
Lemma isIn_singleproduct (a : A) (b : B) (c : A) : forall (Y : FSet B),
isIn (a, b) (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (isIn b Y).
Proof.
intros a b c.
hinduction ; try (intros ; apply path_ishprop).
- apply path_hprop. symmetry. apply prod_empty_r.
- apply path_hprop ; symmetry ; apply prod_empty_r.
- intros d.
apply path_iff_hprop.
* intros.
strip_truncations.
split ; apply tr ; try (apply (ap fst X)) ; try (apply (ap snd X)).
strip_truncations.
split ; apply tr ; try (apply (ap fst X)) ; try (apply (ap snd X)).
* intros [Z1 Z2].
strip_truncations.
rewrite Z1, Z2.
apply (tr idpath).
strip_truncations.
rewrite Z1, Z2.
apply (tr idpath).
- intros X1 X2 HX1 HX2.
unfold lor.
apply path_iff_hprop.
* intros X.
strip_truncations.
destruct X as [H1 | H1].
** rewrite HX1 in H1.
destruct H1 as [H1 H2].
split.
destruct X as [H1 | H1] ; rewrite ?HX1, ?HX2 in H1 ; destruct H1 as [H1 H2].
** split.
*** apply H1.
*** apply (tr(inl H2)).
** rewrite HX2 in H1.
destruct H1 as [H1 H2].
split.
** split.
*** apply H1.
*** apply (tr(inr H2)).
* intros [H1 H2].
@@ -176,15 +170,14 @@ Section properties.
rewrite HX1, HX2.
destruct H2 as [Hb1 | Hb2].
** left.
split ; try (apply (tr H1)) ; try (apply Hb1).
split ; try (apply (tr H1)) ; try (apply Hb1).
** right.
split ; try (apply (tr H1)) ; try (apply Hb2).
Defined.
Definition isIn_product : forall (a : A) (b : B) (X : FSet A) (Y : FSet B),
Definition isIn_product (a : A) (b : B) (X : FSet A) (Y : FSet B) :
isIn (a,b) (product X Y) = land (isIn a X) (isIn b Y).
Proof.
intros a b X Y.
hinduction X ; try (intros ; apply path_ishprop).
- apply path_hprop ; symmetry ; apply prod_empty_l.
- intros.
@@ -194,18 +187,14 @@ Section properties.
apply path_iff_hprop.
* intros X.
strip_truncations.
destruct X as [[H3 H4] | [H3 H4]].
** split.
*** apply (tr(inl H3)).
*** apply H4.
** split.
*** apply (tr(inr H3)).
*** apply H4.
destruct X as [[H3 H4] | [H3 H4]] ; split ; try (apply H4).
** apply (tr(inl H3)).
** apply (tr(inr H3)).
* intros [H1 H2].
strip_truncations.
destruct H1 as [H1 | H1].
** apply tr ; left ; split ; assumption.
** apply tr ; right ; split ; assumption.
destruct H1 as [H1 | H1] ; apply tr.
** left ; split ; assumption.
** right ; split ; assumption.
Defined.
(* The proof can be simplified using extensionality *)