mirror of https://github.com/nmvdw/HITs-Examples
Added idempotency of the intersection
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Niels
parent
13737556c6
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6bb5e8b690
249
FSet.v
249
FSet.v
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@ -347,14 +347,13 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2)
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- apply nl.
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- apply idem.
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- intros x y P Q.
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rewrite assoc.
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rewrite (comm A x y).
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rewrite <- (assoc A y x x).
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rewrite P.
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rewrite (comm A y x).
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rewrite <- (assoc A x y y).
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rewrite Q.
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reflexivity.
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etransitivity. apply assoc.
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etransitivity. apply (ap (fun p => U (U p x) y) (comm A x y)).
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etransitivity. apply (ap (fun p => U p y) (assoc A y x x))^.
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etransitivity. apply (ap (fun p => U (U y p) y) P).
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etransitivity. apply (ap (fun p => U p y) (comm A y x)).
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etransitivity. apply (assoc A x y y)^.
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apply (ap (fun p => U x p) Q).
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Defined.
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Definition isIn : A -> FSet A -> Bool.
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@ -371,8 +370,6 @@ simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- intros a'. destruct (deceq a a'); reflexivity.
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Defined.
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Set Implicit Arguments.
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Definition comprehension :
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(A -> Bool) -> FSet A -> FSet A.
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Proof.
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@ -382,63 +379,16 @@ simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- intro a.
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refine (if (P a) then L a else E).
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- apply U.
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- intros. cbv. apply assoc.
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- intros. cbv. apply comm.
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- intros. cbv. apply nl.
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- intros. cbv. apply nr.
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- intros. cbv.
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destruct (P x); simpl.
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- intros. apply assoc.
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- intros. apply comm.
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- intros. apply nl.
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- intros. apply nr.
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- intros. cbn.
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destruct (P x).
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+ apply idem.
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+ apply nl.
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Defined.
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Definition intersection :
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FSet A -> FSet A -> FSet A.
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Proof.
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intros X Y.
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apply (comprehension (fun (a : A) => isIn a X) Y).
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Defined.
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Lemma intersection_E : forall x,
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intersection E x = E.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- reflexivity.
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- intro a.
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reflexivity.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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apply nl.
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Defined.
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Theorem intersection_La : forall a x,
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intersection (L a) x = if isIn a x then L a else E.
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Proof.
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intro a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- reflexivity.
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- intro b.
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cbn.
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rewrite deceq_sym.
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unfold deceq.
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destruct (dec (a = b)).
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* rewrite p ; reflexivity.
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* reflexivity.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y).
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* apply idem.
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* apply nr.
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* apply nl.
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* apply nl.
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Defined.
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Theorem comprehension_or : forall ϕ ψ x,
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comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x).
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Proof.
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@ -463,13 +413,168 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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reflexivity.
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Defined.
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Definition intersection (X Y : FSet A) : FSet A := comprehension (fun (a : A) => isIn a X) Y.
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Theorem intersection_idem : forall x, intersection x x = x.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
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- reflexivity.
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- intro a.
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unfold deceq.
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destruct (dec (a = a)).
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* reflexivity.
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* pose (n idpath) ; contradiction.
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- intros x y P Q.
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rewrite comprehension_or.
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rewrite comprehension_or.
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unfold intersection in P.
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unfold intersection in Q.
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rewrite P.
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rewrite Q.
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Admitted.
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Theorem intersection_EX : forall x,
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intersection E x = E.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- reflexivity.
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- intro a.
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reflexivity.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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apply nl.
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Defined.
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Definition intersection_XE x : intersection x E = E := idpath.
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Theorem intersection_La : forall a x,
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intersection (L a) x = if isIn a x then L a else E.
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Proof.
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intro a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- reflexivity.
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- intro b.
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cbn.
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rewrite deceq_sym.
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unfold deceq.
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destruct (dec (a = b)).
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* rewrite p ; reflexivity.
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* reflexivity.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y).
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* apply idem.
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* apply nr.
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* apply nl.
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* apply nl.
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Defined.
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(*
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Theorem intersection_comm : forall x y,
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intersection x y = intersection y x.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- intros.
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rewrite intersection_E.
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reflexivity.
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- intros a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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* reflexivity.
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* intro b.
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admit.
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* intros x y.
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destruct (isIn a x) ; destruct (isIn a y) ; intros P Q.
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+ rewrite P.
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*)
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Theorem comp_false : forall x,
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comprehension (fun _ => false) x = E.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
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- reflexivity.
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- intro a ; reflexivity.
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- intros x y P Q.
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rewrite P.
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rewrite Q.
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apply nl.
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Defined.
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(*Theorem union_dist : forall x y z,
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intersection z (U x y) = U (intersection z x) (intersection z y).
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Proof.
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intros x y.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- rewrite intersection_E.
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rewrite intersection_E.
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rewrite comp_false.
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rewrite comp_false.
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reflexivity.
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- intro a.
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*)
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Theorem union_dist : forall x y z,
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intersection (U x y) z = U (intersection x z) (intersection y z).
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- intros.
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rewrite nl.
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rewrite intersection_EX.
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rewrite nl.
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reflexivity.
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- intro a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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* intros.
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rewrite nr.
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rewrite intersection_EX.
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rewrite nr.
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reflexivity.
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* intros b.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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+ rewrite nl. reflexivity.
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+ intros c.
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unfold deceq.
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destruct (dec (c = a)) ; destruct (dec (c = b)) ; cbn.
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{ rewrite idem. reflexivity. }
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{ rewrite nr. reflexivity. }
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{ rewrite nl. reflexivity. }
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{ rewrite nl. reflexivity. }
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+ intros x y P Q.
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rewrite comprehension_or.
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rewrite comprehension_or.
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rewrite assoc.
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rewrite <- (assoc A (comprehension (fun a0 : A => deceq a0 a) x) (comprehension (fun a0 : A => deceq a0 b) x)).
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rewrite (comm A (comprehension (fun a0 : A => deceq a0 b) x) (comprehension (fun a0 : A => deceq a0 a) y)).
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rewrite assoc.
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rewrite <- assoc.
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reflexivity.
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+ admit.
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+ admit.
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+ admit.
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+ admit.
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+ admit.
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* intros x y P Q z.
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cbn.
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Admitted.
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Theorem intersection_isIn : forall a x y,
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isIn a (intersection x y) = andb (isIn a x) (isIn a y).
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Proof.
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intros a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- intros y.
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rewrite intersection_E.
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rewrite intersection_EX.
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reflexivity.
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- intros b y.
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rewrite intersection_La.
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@ -491,22 +596,20 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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{ reflexivity. }
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+ reflexivity.
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- intros x y P Q z.
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enough (intersection (U x y) z = U (intersection x z) (intersection y z)).
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rewrite X.
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rewrite union_dist.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y) ; destruct (isIn a z) ; reflexivity.
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admit.
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Admitted.
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Theorem intersection_assoc : forall x y z,
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Theorem intersection_assoc x y z :
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intersection x (intersection y z) = intersection (intersection x y) z.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- intros y z.
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cbn.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ x) ; cbn ; try (intros ; apply set_path2).
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- intros.
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exact _.
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- cbn.
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rewrite intersection_E.
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rewrite intersection_E.
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rewrite intersection_E.
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@ -532,4 +635,4 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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cbn.
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rewrite comprehension_or.
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reflexivity.
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Admitted.
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@ -267,5 +267,48 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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reflexivity.
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*)
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End properties.
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Theorem comprehension_subset : forall ϕ (X : FSet A),
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U (comprehension ϕ X) X = X.
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Proof.
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intros ϕ.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
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- apply nl.
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- intro a.
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destruct (ϕ a).
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* apply union_idem.
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* apply nl.
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- intros X Y P Q.
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rewrite assoc.
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rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
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rewrite (comm (comprehension ϕ Y) X).
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rewrite assoc.
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rewrite P.
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rewrite <- assoc.
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rewrite Q.
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reflexivity.
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Defined.
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Theorem intersection_idem : forall (X : FSet A), intersection X X = X.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
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- reflexivity.
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- intro a.
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unfold operations.deceq.
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destruct (dec (a = a)).
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* reflexivity.
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* contradiction (n idpath).
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- intros X Y IHX IHY.
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cbn in *.
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rewrite comprehension_or.
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rewrite comprehension_or.
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unfold intersection in *.
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rewrite IHX.
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rewrite IHY.
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rewrite comprehension_subset.
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rewrite (comm X).
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rewrite comprehension_subset.
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reflexivity.
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Defined.
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End properties.
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