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Changed names, further work on distributive laws

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Niels
2017-05-24 11:47:30 +02:00
parent e63f1b8bf5
commit 565fecec30
1 changed files with 95 additions and 3 deletions
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96
FiniteSets/properties.v
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@@ -151,7 +151,7 @@ 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). Lemma distributive_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. Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn. simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- symmetry ; apply nl. - symmetry ; apply nl.
@@ -175,7 +175,7 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2)
reflexivity. reflexivity.
Defined. Defined.
Theorem absorbtion_1 (X1 X2 Y : FSet A) : Theorem distributive_intersection_U (X1 X2 Y : FSet A) :
intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y). intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
Proof. Proof.
simple refine (FSet_ind A simple refine (FSet_ind A
@@ -373,5 +373,97 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2)
reflexivity. reflexivity.
Defined. Defined.
Theorem comprehension_all : forall (X : FSet A),
comprehension (fun a => isIn a X) X = X.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- reflexivity.
- intro a.
unfold operations.deceq.
destruct (dec (a = a)).
* reflexivity.
* contradiction (n idpath).
- intros X1 X2 P Q.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite P.
rewrite Q.
rewrite comprehension_subset.
rewrite (comm X1).
rewrite comprehension_subset.
reflexivity.
Defined.
Theorem distributive_U_int (X1 X2 Y : FSet A) :
U (intersection X1 X2) Y = intersection (U X1 Y) (U X2 Y).
Proof.
simple refine (FSet_ind A
(fun z => U (intersection z X2) Y = intersection (U z Y) (U X2 Y))
_ _ _ _ _ _ _ _ _ X1) ; try (intros ; apply set_path2) ; cbn.
- rewrite intersection_0l.
rewrite nl.
rewrite comprehension_all.
pose (intersection_comm Y X2).
unfold intersection in p.
rewrite p.
rewrite comprehension_subset.
reflexivity.
- intros.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite intersection_La.
unfold operations.deceq.
admit.
- unfold intersection.
cbn.
intros Z1 Z2 P Q.
rewrite comprehension_or.
assert (U (U (comprehension (fun a : A => isIn a Z1) X2) (comprehension (fun a : A => isIn a Z2) X2))
Y = U (U (comprehension (fun a : A => isIn a Z1) X2) (comprehension (fun a : A => isIn a Z2) X2))
(U Y Y)).
rewrite (union_idem Y).
reflexivity.
rewrite X.
rewrite <- assoc.
rewrite (assoc (comprehension (fun a : A => isIn a Z2) X2)).
rewrite Q.
rewrite (comm (U (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) Y).
rewrite assoc.
rewrite P.
rewrite <- assoc.
rewrite (assoc (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)).
rewrite (comm (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)).
rewrite <- assoc.
rewrite assoc.
enough (C : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) X2)
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)) = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) X2)).
rewrite C.
enough (D : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) = (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) Y)).
rewrite D.
reflexivity.
* rewrite comprehension_or.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite <- assoc.
rewrite (comm (comprehension (fun a : A => isIn a Y) Y)).
rewrite <- (assoc (comprehension (fun a : A => isIn a Z2) Y)).
rewrite union_idem.
rewrite assoc.
reflexivity.
* rewrite comprehension_or.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite <- assoc.
rewrite (comm (comprehension (fun a : A => isIn a Y) X2)).
rewrite <- (assoc (comprehension (fun a : A => isIn a Z2) X2)).
rewrite union_idem.
rewrite assoc.
reflexivity.
Admitted.
End properties. End properties.