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Niels
2017-08-07 22:13:42 +02:00
parent 8c10ab1c0c
commit e498b93f16
2 changed files with 72 additions and 1 deletions
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30
FiniteSets/fsets/properties.v
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@@ -137,6 +137,36 @@ Section properties.
** intros ; apply tr ; right ; assumption.
Defined.
Context {B : Type}.
Definition isIn_product : forall (a : A) (b : B) (X : FSet A) (Y : FSet B),
isIn a X -> isIn b Y -> isIn (a,b) (product X Y).
Proof.
intros a b X Y.
hinduction X ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
- contradiction.
- intros c Hc.
hinduction Y ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
* contradiction.
* intros d Hd.
strip_truncations.
apply tr.
rewrite Hc, Hd.
reflexivity.
* intros Y1 Y2 HY1 HY2 HOr.
strip_truncations.
apply tr.
destruct HOr as [H1 | H2].
** apply (inl (HY1 H1)).
** apply (inr (HY2 H2)).
- intros X1 X2 HX1 HX2 Hor HY.
strip_truncations.
apply tr.
destruct Hor as [H1 | H2].
* apply (inl(HX1 H1 HY)).
* apply (inr (HX2 H2 HY)).
Defined.
(* The proof can be simplified using extensionality *)
(** comprehension properties *)
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.