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HITs-Examples
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Added idempotency of the intersection

This commit is contained in:
Niels 2017-05-23 21:31:45 +02:00
parent 13737556c6
commit 6bb5e8b690
2 changed files with 221 additions and 75 deletions
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249
FSet.v
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@ -347,14 +347,13 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2)
- apply nl. - apply nl.
- apply idem. - apply idem.
- intros x y P Q. - intros x y P Q.
rewrite assoc. etransitivity. apply assoc.
rewrite (comm A x y). etransitivity. apply (ap (fun p => U (U p x) y) (comm A x y)).
rewrite <- (assoc A y x x). etransitivity. apply (ap (fun p => U p y) (assoc A y x x))^.
rewrite P. etransitivity. apply (ap (fun p => U (U y p) y) P).
rewrite (comm A y x). etransitivity. apply (ap (fun p => U p y) (comm A y x)).
rewrite <- (assoc A x y y). etransitivity. apply (assoc A x y y)^.
rewrite Q. apply (ap (fun p => U x p) Q).
reflexivity.
Defined. Defined.
Definition isIn : A -> FSet A -> Bool. Definition isIn : A -> FSet A -> Bool.
@ -371,8 +370,6 @@ simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
- intros a'. destruct (deceq a a'); reflexivity. - intros a'. destruct (deceq a a'); reflexivity.
Defined. Defined.
Set Implicit Arguments.
Definition comprehension : Definition comprehension :
(A -> Bool) -> FSet A -> FSet A. (A -> Bool) -> FSet A -> FSet A.
Proof. Proof.
@ -382,63 +379,16 @@ simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
- intro a. - intro a.
refine (if (P a) then L a else E). refine (if (P a) then L a else E).
- apply U. - apply U.
- intros. cbv. apply assoc. - intros. apply assoc.
- intros. cbv. apply comm. - intros. apply comm.
- intros. cbv. apply nl. - intros. apply nl.
- intros. cbv. apply nr. - intros. apply nr.
- intros. cbv. - intros. cbn.
destruct (P x); simpl. destruct (P x).
+ apply idem. + apply idem.
+ apply nl. + apply nl.
Defined. Defined.
Definition intersection :
FSet A -> FSet A -> FSet A.
Proof.
intros X Y.
apply (comprehension (fun (a : A) => isIn a X) Y).
Defined.
Lemma intersection_E : forall x,
intersection E x = E.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
- reflexivity.
- intro a.
reflexivity.
- unfold intersection.
intros x y P Q.
cbn.
rewrite P.
rewrite Q.
apply nl.
Defined.
Theorem intersection_La : forall a x,
intersection (L a) x = if isIn a x then L a else E.
Proof.
intro a.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
- reflexivity.
- intro b.
cbn.
rewrite deceq_sym.
unfold deceq.
destruct (dec (a = b)).
* rewrite p ; reflexivity.
* reflexivity.
- unfold intersection.
intros x y P Q.
cbn.
rewrite P.
rewrite Q.
destruct (isIn a x) ; destruct (isIn a y).
* apply idem.
* apply nr.
* apply nl.
* apply nl.
Defined.
Theorem comprehension_or : forall ϕ ψ x, Theorem comprehension_or : forall ϕ ψ x,
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x). comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x).
Proof. Proof.
@ -463,13 +413,168 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
reflexivity. reflexivity.
Defined. Defined.
Definition intersection (X Y : FSet A) : FSet A := comprehension (fun (a : A) => isIn a X) Y.
Theorem intersection_idem : forall x, intersection x x = x.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- reflexivity.
- intro a.
unfold deceq.
destruct (dec (a = a)).
* reflexivity.
* pose (n idpath) ; contradiction.
- intros x y P Q.
rewrite comprehension_or.
rewrite comprehension_or.
unfold intersection in P.
unfold intersection in Q.
rewrite P.
rewrite Q.
Admitted.
Theorem intersection_EX : forall x,
intersection E x = E.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
- reflexivity.
- intro a.
reflexivity.
- unfold intersection.
intros x y P Q.
cbn.
rewrite P.
rewrite Q.
apply nl.
Defined.
Definition intersection_XE x : intersection x E = E := idpath.
Theorem intersection_La : forall a x,
intersection (L a) x = if isIn a x then L a else E.
Proof.
intro a.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
- reflexivity.
- intro b.
cbn.
rewrite deceq_sym.
unfold deceq.
destruct (dec (a = b)).
* rewrite p ; reflexivity.
* reflexivity.
- unfold intersection.
intros x y P Q.
cbn.
rewrite P.
rewrite Q.
destruct (isIn a x) ; destruct (isIn a y).
* apply idem.
* apply nr.
* apply nl.
* apply nl.
Defined.
(*
Theorem intersection_comm : forall x y,
intersection x y = intersection y x.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
- intros.
rewrite intersection_E.
reflexivity.
- intros a.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
* reflexivity.
* intro b.
admit.
* intros x y.
destruct (isIn a x) ; destruct (isIn a y) ; intros P Q.
+ rewrite P.
*)
Theorem comp_false : forall x,
comprehension (fun _ => false) x = E.
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- reflexivity.
- intro a ; reflexivity.
- intros x y P Q.
rewrite P.
rewrite Q.
apply nl.
Defined.
(*Theorem union_dist : forall x y z,
intersection z (U x y) = U (intersection z x) (intersection z y).
Proof.
intros x y.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
- rewrite intersection_E.
rewrite intersection_E.
rewrite comp_false.
rewrite comp_false.
reflexivity.
- intro a.
*)
Theorem union_dist : forall x y z,
intersection (U x y) z = U (intersection x z) (intersection y z).
Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
- intros.
rewrite nl.
rewrite intersection_EX.
rewrite nl.
reflexivity.
- intro a.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
* intros.
rewrite nr.
rewrite intersection_EX.
rewrite nr.
reflexivity.
* intros b.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
+ rewrite nl. reflexivity.
+ intros c.
unfold deceq.
destruct (dec (c = a)) ; destruct (dec (c = b)) ; cbn.
{ rewrite idem. reflexivity. }
{ rewrite nr. reflexivity. }
{ rewrite nl. reflexivity. }
{ rewrite nl. reflexivity. }
+ intros x y P Q.
rewrite comprehension_or.
rewrite comprehension_or.
rewrite assoc.
rewrite <- (assoc A (comprehension (fun a0 : A => deceq a0 a) x) (comprehension (fun a0 : A => deceq a0 b) x)).
rewrite (comm A (comprehension (fun a0 : A => deceq a0 b) x) (comprehension (fun a0 : A => deceq a0 a) y)).
rewrite assoc.
rewrite <- assoc.
reflexivity.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
* intros x y P Q z.
cbn.
Admitted.
Theorem intersection_isIn : forall a x y, Theorem intersection_isIn : forall a x y,
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.
intros a. intros a.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn. simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
- intros y. - intros y.
rewrite intersection_E. rewrite intersection_EX.
reflexivity. reflexivity.
- intros b y. - intros b y.
rewrite intersection_La. rewrite intersection_La.
@ -491,22 +596,20 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
{ reflexivity. } { reflexivity. }
+ reflexivity. + reflexivity.
- intros x y P Q z. - intros x y P Q z.
enough (intersection (U x y) z = U (intersection x z) (intersection y z)). rewrite union_dist.
rewrite X.
cbn. cbn.
rewrite P. rewrite P.
rewrite Q. rewrite Q.
destruct (isIn a x) ; destruct (isIn a y) ; destruct (isIn a z) ; reflexivity. destruct (isIn a x) ; destruct (isIn a y) ; destruct (isIn a z) ; reflexivity.
admit.
Admitted. Admitted.
Theorem intersection_assoc x y z :
Theorem intersection_assoc : forall x y z,
intersection x (intersection y z) = intersection (intersection x y) z. intersection x (intersection y z) = intersection (intersection x y) z.
Proof. Proof.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2). simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ x) ; cbn ; try (intros ; apply set_path2).
- intros y z. - intros.
cbn. exact _.
- cbn.
rewrite intersection_E. rewrite intersection_E.
rewrite intersection_E. rewrite intersection_E.
rewrite intersection_E. rewrite intersection_E.
@ -532,4 +635,4 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
cbn. cbn.
rewrite comprehension_or. rewrite comprehension_or.
reflexivity. reflexivity.
Admitted.

45
FiniteSets/properties.v
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@ -267,5 +267,48 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
reflexivity. reflexivity.
*) *)
End properties. Theorem comprehension_subset : forall ϕ (X : FSet A),
U (comprehension ϕ X) X = X.
Proof.
intros ϕ.
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
- apply nl.
- intro a.
destruct (ϕ a).
* apply union_idem.
* apply nl.
- intros X Y P Q.
rewrite assoc.
rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
rewrite (comm (comprehension ϕ Y) X).
rewrite assoc.
rewrite P.
rewrite <- assoc.
rewrite Q.
reflexivity.
Defined.
Theorem intersection_idem : forall (X : FSet A), intersection 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 X Y IHX IHY.
cbn in *.
rewrite comprehension_or.
rewrite comprehension_or.
unfold intersection in *.
rewrite IHX.
rewrite IHY.
rewrite comprehension_subset.
rewrite (comm X).
rewrite comprehension_subset.
reflexivity.
Defined.
End properties.