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Added membership of product

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This commit is contained in:
Niels
2017-08-07 23:15:25 +02:00
parent e498b93f16
commit 30004e1c8b
2 changed files with 79 additions and 33 deletions
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27
FiniteSets/fsets/operations.v
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@@ -92,23 +92,28 @@ Section operations.
Defined.
Context {B : Type}.
Definition single_product (a : A) : FSet B -> FSet (A * B).
Proof.
hrecursion.
- apply ∅.
- intro b.
apply {|(a, b)|}.
- apply U.
- intros X Y Z ; apply assoc.
- intros X Y ; apply comm.
- intros ; apply nl.
- intros ; apply nr.
- intros ; apply idem.
Defined.
Definition product : FSet A -> FSet B -> FSet (A * B).
Proof.
intros X Y.
hrecursion X.
- apply ∅.
- intro a.
hrecursion Y ; simpl in *.
* apply ∅.
* intro b.
apply {|(a, b)|}.
* apply U.
* intros X Y Z ; apply assoc.
* intros X Y ; apply comm.
* intros ; apply nl.
* intros ; apply nr.
* intros ; apply idem.
apply (single_product a Y).
- apply U.
- intros ; apply assoc.
- intros ; apply comm.

85
FiniteSets/fsets/properties.v
View File

@@ -138,33 +138,74 @@ Section properties.
Defined.
Context {B : Type}.
Lemma isIn_singleproduct : forall (a : A) (b : B) (c : A) (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.
- intros d.
apply path_iff_hprop.
* intros.
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).
- 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.
*** apply H1.
*** apply (tr(inl H2)).
** rewrite HX2 in H1.
destruct H1 as [H1 H2].
split.
*** apply H1.
*** apply (tr(inr H2)).
* intros [H1 H2].
strip_truncations.
apply tr.
rewrite HX1, HX2.
destruct H2 as [Hb1 | Hb2].
** left.
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),
isIn a X -> isIn b Y -> isIn (a,b) (product X Y).
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_forall ; intro ; apply path_ishprop).
- contradiction.
- intros c Hc.
hinduction Y ; try (intros ; apply path_forall ; intro ; apply path_ishprop).
* contradiction.
* intros d Hd.
hinduction X ; try (intros ; apply path_ishprop).
- apply path_hprop ; symmetry ; apply prod_empty_l.
- intros.
apply isIn_singleproduct.
- intros X1 X2 HX1 HX2.
rewrite HX1, HX2.
apply path_iff_hprop.
* intros X.
strip_truncations.
apply tr.
rewrite Hc, Hd.
reflexivity.
* intros Y1 Y2 HY1 HY2 HOr.
destruct X as [[H3 H4] | [H3 H4]].
** split.
*** apply (tr(inl H3)).
*** apply H4.
** split.
*** apply (tr(inr H3)).
*** apply H4.
* intros [H1 H2].
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)).
destruct H1 as [H1 | H1].
** apply tr ; left ; split ; assumption.
** apply tr ; right ; split ; assumption.
Defined.
(* The proof can be simplified using extensionality *)