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More absorbtion

This commit is contained in:
Niels 2017-05-24 12:10:39 +02:00
parent 565fecec30
commit 6ffbbb440f
1 changed files with 50 additions and 3 deletions
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53
FiniteSets/properties.v
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@ -187,7 +187,7 @@ simple refine (FSet_ind A
reflexivity.
- intro a.
rewrite intersection_La.
rewrite absorb_La.
rewrite distributive_La.
rewrite intersection_La.
reflexivity.
- intros Z1 Z2 P Q.
@ -225,7 +225,7 @@ simple refine (FSet_ind A (fun z => isIn a (intersection z y) = andb (isIn a z)
{ reflexivity. }
+ reflexivity.
- intros X1 X2 P Q.
rewrite absorbtion_1.
rewrite distributive_intersection_U.
cbn.
rewrite P.
rewrite Q.
@ -444,7 +444,7 @@ simple refine (FSet_ind A
(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.
* repeat (rewrite comprehension_or).
rewrite comprehension_or.
rewrite comprehension_or.
rewrite comprehension_or.
@ -466,4 +466,51 @@ simple refine (FSet_ind A
reflexivity.
Admitted.
Theorem absorb_0 (X Y : FSet A) : U X (intersection X Y) = X.
Proof.
simple refine (FSet_ind A
(fun z => U z (intersection z Y) = z)
_ _ _ _ _ _ _ _ _ X) ; try (intros ; apply set_path2) ; cbn.
- rewrite nl.
apply intersection_0l.
- intro a.
rewrite intersection_La.
destruct (isIn a Y).
* apply union_idem.
* apply nr.
- intros X1 X2 P Q.
rewrite distributive_intersection_U.
rewrite <- assoc.
rewrite (comm X2).
rewrite assoc.
rewrite assoc.
rewrite P.
rewrite <- assoc.
rewrite (comm _ X2).
rewrite Q.
reflexivity.
Defined.
Theorem absorb_1 (X Y : FSet A) : intersection X (U X Y) = X.
Proof.
simple refine (FSet_ind A
(fun z => intersection z (U z Y) = z)
_ _ _ _ _ _ _ _ _ X) ; try (intros ; apply set_path2).
- cbn.
rewrite nl.
apply comprehension_false.
- intro a.
simpl.
unfold operations.deceq.
rewrite intersection_La.
destruct (dec (a = a)).
* destruct (isIn a Y).
+ apply union_idem.
+ apply nr.
* contradiction (n idpath).
- intros X1 X2 P Q.
cbn in *.
symmetry.
rewrite <- P.
rewrite <- Q.
End properties.