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One absorption law

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Niels
2017-05-23 23:24:22 +02:00
parent 2f7840c494
commit e63f1b8bf5
1 changed files with 47 additions and 4 deletions
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51
FiniteSets/properties.v
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@@ -126,10 +126,6 @@ Defined.
Definition intersection_0r (X: FSet A): intersection X E = E := idpath. Definition intersection_0r (X: FSet A): intersection X E = E := idpath.
Theorem absorbtion_1 : forall (X1 X2 Y : FSet A),
intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
Admitted.
Theorem intersection_La : forall (a : A) (X : FSet A), Theorem intersection_La : forall (a : A) (X : FSet A),
intersection (L a) X = if isIn a X then L a else E. intersection (L a) X = if isIn a X then L a else E.
Proof. Proof.
@@ -155,6 +151,53 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
* apply nl. * apply nl.
Defined. Defined.
Lemma absorb_La (z : FSet A) (a : A) : forall Y : FSet A, intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- symmetry ; apply nl.
- intros b.
unfold operations.deceq.
destruct (dec (b = a)) ; cbn.
* destruct (isIn b z).
+ rewrite union_idem.
reflexivity.
+ rewrite nr.
reflexivity.
* rewrite nl ; reflexivity.
- intros X1 X2 P Q.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite <- assoc.
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X1)).
rewrite <- assoc.
rewrite assoc.
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X2)).
reflexivity.
Defined.
Theorem absorbtion_1 (X1 X2 Y : FSet A) :
intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
Proof.
simple refine (FSet_ind A
(fun z => intersection (U z X2) Y = U (intersection z Y) (intersection X2 Y))
_ _ _ _ _ _ _ _ _ X1) ; try (intros ; apply set_path2) ; cbn.
- rewrite intersection_0l.
rewrite nl.
rewrite nl.
reflexivity.
- intro a.
rewrite intersection_La.
rewrite absorb_La.
rewrite intersection_La.
reflexivity.
- intros Z1 Z2 P Q.
unfold intersection in *.
cbn.
rewrite comprehension_or.
rewrite comprehension_or.
reflexivity.
Defined.
Theorem intersection_isIn : forall a (x y: FSet A), Theorem intersection_isIn : forall a (x y: FSet A),
isIn a (intersection x y) = andb (isIn a x) (isIn a y). isIn a (intersection x y) = andb (isIn a x) (isIn a y).
Proof. Proof.