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Definition isFilter {X : UU} (F : (X β hProp) β hProp) := isPreFilter F Γ isfilter_notempty F. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isFilter | 38,400 |
Definition Filter (X : UU) := β F : (X β hProp) β hProp, isFilter F. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | Filter | 38,401 |
Definition pr1Filter (X : UU) (F : Filter X) : PreFilter X := pr1 F,, pr1 (pr2 F). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | pr1Filter | 38,402 |
Definition make_Filter {X : UU} (F : (X β hProp) β hProp) (Himp : isfilter_imply F) (Htrue : isfilter_htrue F) (Hand : isfilter_and F) (Hempty : isfilter_notempty F) : Filter X := F ,, (Himp ,, (isfilter_finite_intersection_carac F Htrue Hand)) ,, Hempty. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | make_Filter | 38,403 |
Lemma emptynofilter : β F : (empty β hProp) β hProp, Β¬ isFilter F. Proof. intros F Hf. generalize (isfilter_finite_intersection_htrue _ (pr2 (pr1 Hf))) ; intros Htrue. generalize (pr2 Hf _ Htrue). apply factor_through_squash. - apply isapropempty. - intros x. apply (pr1 x). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | emptynofilter | 38,404 |
Lemma filter_imply : isfilter_imply F. Proof. exact (pr1 (pr2 F)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_imply | 38,405 |
Lemma filter_finite_intersection : isfilter_finite_intersection F. Proof. exact (pr2 (pr2 F)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_finite_intersection | 38,406 |
Lemma filter_htrue : isfilter_htrue F. Proof. apply isfilter_finite_intersection_htrue. exact filter_finite_intersection. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_htrue | 38,407 |
Lemma filter_and : isfilter_and F. Proof. apply isfilter_finite_intersection_and. exact filter_finite_intersection. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_and | 38,408 |
Lemma filter_forall : β A : X β hProp, (β x : X, A x) β F A. Proof. intros A Ha. generalize filter_htrue. apply filter_imply. intros x _. now apply Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_forall | 38,409 |
Lemma filter_notempty : isfilter_notempty F. Proof. exact (pr2 (pr2 F)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_notempty | 38,410 |
Lemma filter_const : β A : hProp, F (Ξ» _ : X, A) β Β¬ (Β¬ A). Proof. intros A Fa Ha. generalize (filter_notempty _ Fa). apply factor_through_squash. - apply isapropempty. - intros x ; generalize (pr2 x); clear x. exact Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_const | 38,411 |
Lemma isasetPreFilter (X : UU) : isaset (PreFilter X). Proof. simple refine (isaset_carrier_subset (make_hSet _ _) (Ξ» _, make_hProp _ _)). - apply impred_isaset ; intros _. apply isasethProp. - apply isapropdirprod. + apply isaprop_isfilter_imply. + apply isaprop_isfilter_finite_intersection. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isasetPreFilter | 38,412 |
Lemma isasetFilter (X : UU) : isaset (Filter X). Proof. simple refine (isaset_carrier_subset (make_hSet _ _) (Ξ» _, make_hProp _ _)). - apply impred_isaset ; intros _. apply isasethProp. - apply isapropdirprod. + apply isapropdirprod. * apply isaprop_isfilter_imply. * apply isaprop_isfilter_finite_intersection. + apply isaprop_isfilter_notempty. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isasetFilter | 38,413 |
Definition filter_le {X : UU} (F G : PreFilter X) := β A : X β hProp, G A β F A. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filter_le | 38,414 |
Lemma istrans_filter_le {X : UU} : β F G H : PreFilter X, filter_le F G β filter_le G H β filter_le F H. Proof. intros F G H Hfg Hgh A Fa. apply Hfg, Hgh, Fa. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | istrans_filter_le | 38,415 |
Lemma isrefl_filter_le {X : UU} : β F : PreFilter X, filter_le F F. Proof. intros F A Fa. exact Fa. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isrefl_filter_le | 38,416 |
Lemma isantisymm_filter_le {X : UU} : β F G : PreFilter X, filter_le F G β filter_le G F β F = G. Proof. intros F G Hle Hge. simple refine (subtypePath_prop (B := Ξ» _, make_hProp _ _) _). - apply isapropdirprod. + apply isaprop_isfilter_imply. + apply isaprop_isfilter_finite_intersection. - apply funextfun ; intros A. apply hPropUnivalence. + now apply Hge. + now apply Hle. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isantisymm_filter_le | 38,417 |
Definition PartialOrder_filter_le (X : UU) : PartialOrder (make_hSet (PreFilter _) (isasetPreFilter X)). Proof. simple refine (make_PartialOrder _ _). - intros F G. simple refine (make_hProp _ _). + apply (filter_le F G). + apply impred_isaprop ; intros A. apply isapropimpl. apply propproperty. - repeat split. + intros F G H ; simpl. apply istrans_filter_le. + intros A ; simpl. apply isrefl_filter_le. + intros F G ; simpl. apply isantisymm_filter_le. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PartialOrder_filter_le | 38,418 |
Definition filterim := Ξ» A : (Y β hProp), F (Ξ» x : X, A (f x)). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterim | 38,419 |
Lemma filterim_imply : isfilter_imply filterim. Proof. intros A B H. apply Himp. intros x. apply H. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterim_imply | 38,420 |
Lemma filterim_htrue : isfilter_htrue filterim. Proof. apply Htrue. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterim_htrue | 38,421 |
Lemma filterim_and : isfilter_and filterim. Proof. intros A B. apply Hand. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterim_and | 38,422 |
Lemma filterim_notempty : isfilter_notempty filterim. Proof. intros A Fa. generalize (Hempty _ Fa). apply hinhfun. intros x. exists (f (pr1 x)). exact (pr2 x). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterim_notempty | 38,423 |
Definition PreFilterIm {X Y : UU} (f : X β Y) (F : PreFilter X) : PreFilter Y. Proof. simple refine (make_PreFilter _ _ _ _). - exact (filterim f F). - apply filterim_imply, filter_imply. - apply filterim_htrue, filter_htrue. - apply filterim_and, filter_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterIm | 38,424 |
Definition FilterIm {X Y : UU} (f : X β Y) (F : Filter X) : Filter Y. Proof. refine (tpair _ _ _). split. - apply (pr2 (PreFilterIm f F)). - apply filterim_notempty, filter_notempty. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterIm | 38,425 |
Lemma PreFilterIm_incr {X Y : UU} : β (f : X β Y) (F G : PreFilter X), filter_le F G β filter_le (PreFilterIm f F) (PreFilterIm f G). Proof. intros f F G Hle A ; simpl. apply Hle. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterIm_incr | 38,426 |
Lemma FilterIm_incr {X Y : UU} : β (f : X β Y) (F G : Filter X), filter_le F G β filter_le (FilterIm f F) (FilterIm f G). Proof. intros f F G Hle A ; simpl. apply Hle. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterIm_incr | 38,427 |
Definition filterlim {X Y : UU} (f : X β Y) (F : PreFilter X) (G : PreFilter Y) := filter_le (PreFilterIm f F) G. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterlim | 38,428 |
Lemma filterlim_comp {X Y Z : UU} : β (f : X β Y) (g : Y β Z) (F : PreFilter X) (G : PreFilter Y) (H : PreFilter Z), filterlim f F G β filterlim g G H β filterlim (funcomp f g) F H. Proof. intros f g F G H Hf Hg A Fa. specialize (Hg _ Fa). specialize (Hf _ Hg). apply Hf. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterlim_comp | 38,429 |
Lemma filterlim_decr_1 {X Y : UU} : β (f : X β Y) (F F' : PreFilter X) (G : PreFilter Y), filter_le F' F β filterlim f F G β filterlim f F' G. Proof. intros f F F' G Hf Hle A Ha. specialize (Hle _ Ha). specialize (Hf _ Hle). apply Hf. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterlim_decr_1 | 38,430 |
Lemma filterlim_incr_2 {X Y : UU} : β (f : X β Y) (F : PreFilter X) (G G' : PreFilter Y), filter_le G G' β filterlim f F G β filterlim f F G'. Proof. intros f F G G' Hg Hle A Ha. specialize (Hg _ Ha). specialize (Hle _ Hg). exact Hle. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterlim_incr_2 | 38,431 |
Definition filterdom : (X β hProp) β hProp := Ξ» A : X β hProp, F (Ξ» x : X, make_hProp (dom x β A x) (isapropimpl _ _ (propproperty _))). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdom | 38,432 |
Lemma filterdom_imply : isfilter_imply filterdom. Proof. intros A B Himpl. apply Himp. intros x Ax Hx. apply Himpl, Ax, Hx. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdom_imply | 38,433 |
Lemma filterdom_htrue : isfilter_htrue filterdom. Proof. apply Himp with (2 := Htrue). intros x H _. exact H. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdom_htrue | 38,434 |
Lemma filterdom_and : isfilter_and filterdom. Proof. intros A B Ha Hb. generalize (Hand _ _ Ha Hb). apply Himp. intros x ABx Hx. split. - apply (pr1 ABx), Hx. - apply (pr2 ABx), Hx. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdom_and | 38,435 |
Lemma filterdom_notempty : isfilter_notempty filterdom. Proof. intros. intros A Fa. generalize (Hdom _ Fa). apply hinhfun. intros x. exists (pr1 x). apply (pr2 (pr2 x)), (pr1 (pr2 x)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdom_notempty | 38,436 |
Definition PreFilterDom {X : UU} (F : PreFilter X) (dom : X β hProp) : PreFilter X. Proof. simple refine (make_PreFilter _ _ _ _). - exact (filterdom F dom). - apply filterdom_imply, filter_imply. - apply filterdom_htrue. + apply filter_imply. + apply filter_htrue. - apply filterdom_and. + apply filter_imply. + apply filter_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterDom | 38,437 |
Definition FilterDom {X : UU} (F : Filter X) (dom : X β hProp) (Hdom : β P, F P β β x, dom x β§ P x) : Filter X. Proof. refine (tpair _ _ _). split. - apply (pr2 (PreFilterDom F dom)). - apply filterdom_notempty. exact Hdom. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterDom | 38,438 |
Definition filtersubtype : ((β x : X, dom x) β hProp) β hProp := Ξ» A : (β x : X, dom x) β hProp, F (Ξ» x : X, make_hProp (β Hx : dom x, A (x,, Hx)) (impred_isaprop _ (Ξ» _, propproperty _))). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtersubtype | 38,439 |
Lemma filtersubtype_imply : isfilter_imply filtersubtype. Proof. intros A B Himpl. apply Himp. intros x Ax Hx. apply Himpl, Ax. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtersubtype_imply | 38,440 |
Lemma filtersubtype_htrue : isfilter_htrue filtersubtype. Proof. apply Himp with (2 := Htrue). intros x H _. exact H. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtersubtype_htrue | 38,441 |
Lemma filtersubtype_and : isfilter_and filtersubtype. Proof. intros A B Ha Hb. generalize (Hand _ _ Ha Hb). apply Himp. intros x ABx Hx. split. - apply (pr1 ABx). - apply (pr2 ABx). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtersubtype_and | 38,442 |
Lemma filtersubtype_notempty : isfilter_notempty filtersubtype. Proof. intros A Fa. generalize (Hdom _ Fa). apply hinhfun. intros x. exists (pr1 x,,pr1 (pr2 x)). exact (pr2 (pr2 x) (pr1 (pr2 x))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtersubtype_notempty | 38,443 |
Definition PreFilterSubtype {X : UU} (F : PreFilter X) (dom : X β hProp) : PreFilter (β x : X, dom x). Proof. simple refine (make_PreFilter _ _ _ _). - exact (filtersubtype F dom). - apply filtersubtype_imply, filter_imply. - apply filtersubtype_htrue. + apply filter_imply. + apply filter_htrue. - apply filtersubtype_and. + apply filter_imply. + apply filter_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterSubtype | 38,444 |
Definition FilterSubtype {X : UU} (F : Filter X) (dom : X β hProp) (Hdom : β P, F P β β x, dom x β§ P x) : Filter (β x : X, dom x). Proof. refine (tpair _ _ _). split. - apply (pr2 (PreFilterSubtype F dom)). - apply filtersubtype_notempty. exact Hdom. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterSubtype | 38,445 |
Definition filterdirprod : (X Γ Y β hProp) β hProp := Ξ» A : (X Γ Y) β hProp, β (Ax : X β hProp) (Ay : Y β hProp), Fx Ax Γ Fy Ay Γ (β (x : X) (y : Y), Ax x β Ay y β A (x,,y)). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdirprod | 38,446 |
Lemma filterdirprod_imply : isfilter_imply filterdirprod. Proof. intros A B Himpl. apply hinhfun. intros C. generalize (pr1 C) (pr1 (pr2 C)) (pr1 (pr2 (pr2 C))) (pr1 (pr2 (pr2 (pr2 C)))) (pr2 (pr2 (pr2 (pr2 C)))) ; clear C ; intros Ax Ay Fax Fay Ha. exists Ax, Ay. repeat split. + exact Fax. + exact Fay. + intros x y Hx Hy. now apply Himpl, Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdirprod_imply | 38,447 |
Lemma filterdirprod_htrue : isfilter_htrue filterdirprod. Proof. apply hinhpr. exists (Ξ» _:X, htrue), (Ξ» _:Y, htrue). repeat split. + apply Htrue_x. + apply Htrue_y. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdirprod_htrue | 38,448 |
Lemma filterdirprod_and : isfilter_and filterdirprod. Proof. intros A B. apply hinhfun2. intros C D. generalize (pr1 C) (pr1 (pr2 C)) (pr1 (pr2 (pr2 C))) (pr1 (pr2 (pr2 (pr2 C)))) (pr2 (pr2 (pr2 (pr2 C)))) ; clear C ; intros Ax Ay Fax Fay Ha. generalize (pr1 D) (pr1 (pr2 D)) (pr1 (pr2 (pr2 D))) (pr1 (pr2 (pr2 (pr2 D)))) (pr2 (pr2 (pr2 (pr2 D)))) ; clear D ; intros Bx By Fbx Fby Hb. exists (Ξ» x : X, Ax x β§ Bx x), (Ξ» y : Y, Ay y β§ By y). repeat split. + now apply Hand_x. + now apply Hand_y. + apply Ha. * apply (pr1 X0). * apply (pr1 X1). + apply Hb. * apply (pr2 X0). * apply (pr2 X1). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdirprod_and | 38,449 |
Lemma filterdirprod_notempty : isfilter_notempty filterdirprod. Proof. intros A. apply hinhuniv. intros C. generalize (pr1 C) (pr1 (pr2 C)) (pr1 (pr2 (pr2 C))) (pr1 (pr2 (pr2 (pr2 C)))) (pr2 (pr2 (pr2 (pr2 C)))) ; clear C ; intros Ax Ay Fax Fay Ha. generalize (Hempty_x _ Fax) (Hempty_y _ Fay). apply hinhfun2. intros x y. exists (pr1 x,,pr1 y). apply Ha. - exact (pr2 x). - exact (pr2 y). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterdirprod_notempty | 38,450 |
Definition PreFilterDirprod {X Y : UU} (Fx : PreFilter X) (Fy : PreFilter Y) : PreFilter (X Γ Y). Proof. simple refine (make_PreFilter _ _ _ _). - exact (filterdirprod Fx Fy). - apply filterdirprod_imply. - apply filterdirprod_htrue. + apply filter_htrue. + apply filter_htrue. - apply filterdirprod_and. + apply filter_and. + apply filter_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterDirprod | 38,451 |
Definition FilterDirprod {X Y : UU} (Fx : Filter X) (Fy : Filter Y) : Filter (X Γ Y). Proof. refine (tpair _ _ _). split. - apply (pr2 (PreFilterDirprod Fx Fy)). - apply filterdirprod_notempty. + apply filter_notempty. + apply filter_notempty. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterDirprod | 38,452 |
Definition PreFilterPr1 {X Y : UU} (F : PreFilter (X Γ Y)) : PreFilter X := (PreFilterIm pr1 F). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterPr1 | 38,453 |
Definition FilterPr1 {X Y : UU} (F : Filter (X Γ Y)) : Filter X := (FilterIm pr1 F). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterPr1 | 38,454 |
Definition PreFilterPr2 {X Y : UU} (F : PreFilter (X Γ Y)) : PreFilter Y := (PreFilterIm (@pr2 X (Ξ» _ : X, Y)) F). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterPr2 | 38,455 |
Definition FilterPr2 {X Y : UU} (F : Filter (X Γ Y)) : Filter Y := (FilterIm (@pr2 X (Ξ» _ : X, Y)) F). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterPr2 | 38,456 |
Definition filternat : (nat β hProp) β hProp := Ξ» P : nat β hProp, β N : nat, β n : nat, N β€ n β P n. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filternat | 38,457 |
Lemma filternat_imply : isfilter_imply filternat. Proof. intros P Q H. apply hinhfun. intros N. exists (pr1 N). intros n Hn. now apply H, (pr2 N). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filternat_imply | 38,458 |
Lemma filternat_htrue : isfilter_htrue filternat. Proof. apply hinhpr. now exists O. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filternat_htrue | 38,459 |
Lemma filternat_and : isfilter_and filternat. Proof. intros A B. apply hinhfun2. intros Na Nb. exists (max (pr1 Na) (pr1 Nb)). intros n Hn. split. + apply (pr2 Na). eapply istransnatleh, Hn. now apply max_le_l. + apply (pr2 Nb). eapply istransnatleh, Hn. now apply max_le_r. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filternat_and | 38,460 |
Lemma filternat_notempty : isfilter_notempty filternat. Proof. intros A. apply hinhfun. intros N. exists (pr1 N). apply (pr2 N (pr1 N)). now apply isreflnatleh. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filternat_notempty | 38,461 |
Definition FilterNat : Filter nat. Proof. simple refine (make_Filter _ _ _ _ _). - apply filternat. - apply filternat_imply. - apply filternat_htrue. - apply filternat_and. - apply filternat_notempty. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterNat | 38,462 |
Definition filtertop : (X β hProp) β hProp := Ξ» A : X β hProp, make_hProp (β x : X, A x) (impred_isaprop _ (Ξ» _, propproperty _)). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtertop | 38,463 |
Lemma filtertop_imply : isfilter_imply filtertop. Proof. intros A B H Ha x. apply H. simple refine (Ha _). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtertop_imply | 38,464 |
Lemma filtertop_htrue : isfilter_htrue filtertop. Proof. intros x. apply tt. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtertop_htrue | 38,465 |
Lemma filtertop_and : isfilter_and filtertop. Proof. intros A B Ha Hb x. split. + simple refine (Ha _). + simple refine (Hb _). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtertop_and | 38,466 |
Lemma filtertop_notempty : isfilter_notempty filtertop. Proof. intros A Fa. revert x0. apply hinhfun. intros x0. exists x0. simple refine (Fa _). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtertop_notempty | 38,467 |
Definition PreFilterTop {X : UU} : PreFilter X. Proof. simple refine (make_PreFilter _ _ _ _). - exact filtertop. - exact filtertop_imply. - exact filtertop_htrue. - exact filtertop_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterTop | 38,468 |
Definition FilterTop {X : UU} (x0 : β₯ X β₯) : Filter X. Proof. refine (tpair _ _ _). split. - apply (pr2 PreFilterTop). - apply filtertop_notempty, x0. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterTop | 38,469 |
Lemma PreFilterTop_correct {X : UU} : β (F : PreFilter X), filter_le F PreFilterTop. Proof. intros F A Ha. apply filter_forall, Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterTop_correct | 38,470 |
Lemma FilterTop_correct {X : UU} : β (x0 : β₯ X β₯) (F : Filter X), filter_le F (FilterTop x0). Proof. intros x0 F A Ha. apply PreFilterTop_correct, Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterTop_correct | 38,471 |
Definition filterintersection : (X β hProp) β hProp := Ξ» A : X β hProp, make_hProp (β F, FF F β (pr1 F) A) (impred_isaprop _ (Ξ» _, isapropimpl _ _ (propproperty _))). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterintersection | 38,472 |
Lemma filterintersection_imply : isfilter_imply filterintersection. Proof. intros A B H Ha F Hf. apply (Himp F Hf A). - apply H. - simple refine (Ha _ _). exact Hf. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterintersection_imply | 38,473 |
Lemma filterintersection_htrue : isfilter_htrue filterintersection. Proof. intros F Hf. now apply (Htrue F). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterintersection_htrue | 38,474 |
Lemma filterintersection_and : isfilter_and filterintersection. Proof. intros A B Ha Hb F Hf. apply (Hand F Hf). * now simple refine (Ha _ _). * now simple refine (Hb _ _). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterintersection_and | 38,475 |
Lemma filterintersection_notempty : isfilter_notempty filterintersection. Proof. intros A Fa. revert His. apply hinhuniv. intros F. apply (Hempty (pr1 F)). * exact (pr2 F). * simple refine (Fa _ _). exact (pr2 F). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filterintersection_notempty | 38,476 |
Definition PreFilterIntersection {X : UU} (FF : PreFilter X β hProp) : PreFilter X. Proof. intros. simple refine (make_PreFilter _ _ _ _). - apply (filterintersection _ FF). - apply filterintersection_imply. intros F _. apply filter_imply. - apply filterintersection_htrue. intros F _. apply filter_htrue. - apply filterintersection_and. intros F _. apply filter_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterIntersection | 38,477 |
Definition FilterIntersection {X : UU} (FF : Filter X β hProp) (Hff : β F : Filter X, FF F) : Filter X. Proof. simple refine (make_Filter _ _ _ _ _). - apply (filterintersection _ FF). - apply filterintersection_imply. intros F _. apply (filter_imply (pr1Filter _ F)). - apply filterintersection_htrue. intros F _. apply (filter_htrue (pr1Filter _ F)). - apply filterintersection_and. intros F _. apply (filter_and (pr1Filter _ F)). - apply filterintersection_notempty. + intros F _. apply (filter_notempty F). + exact Hff. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterIntersection | 38,478 |
Lemma PreFilterIntersection_glb {X : UU} (FF : PreFilter X β hProp) : (β F : PreFilter X, FF F β filter_le F (PreFilterIntersection FF)) Γ (β F : PreFilter X, (β G : PreFilter X, FF G β filter_le G F) β filter_le (PreFilterIntersection FF) F). Proof. split. - intros F Hf A Ha. now simple refine (Ha _ _). - intros F H A Fa G Hg. apply (H G Hg). apply Fa. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterIntersection_glb | 38,479 |
Lemma FilterIntersection_glb {X : UU} (FF : Filter X β hProp) Hff : (β F : Filter X, FF F β filter_le F (FilterIntersection FF Hff)) Γ (β F : Filter X, (β G : Filter X, FF G β filter_le G F) β filter_le (FilterIntersection FF Hff) F). Proof. split. - intros F Hf A Ha. now simple refine (Ha _ _). - intros F H A Fa G Hg. apply (H G Hg). apply Fa. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterIntersection_glb | 38,480 |
Definition filtergenerated : (X β hProp) β hProp := Ξ» A : X β hProp, β (L' : Sequence (X β hProp)), (β m, L (L' m)) Γ (β x : X, finite_intersection L' x β A x). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtergenerated | 38,481 |
Lemma filtergenerated_imply : isfilter_imply filtergenerated. Proof. intros A B H. apply hinhfun ; intro Ha. exists (pr1 Ha), (pr1 (pr2 Ha)). intros x Hx. apply H. apply (pr2 (pr2 Ha)), Hx. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtergenerated_imply | 38,482 |
Lemma filtergenerated_htrue : isfilter_htrue filtergenerated. Proof. apply hinhpr. exists nil. split. + intros m. induction (nopathsfalsetotrue (pr2 m)). + easy. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtergenerated_htrue | 38,483 |
Lemma filtergenerated_and : isfilter_and filtergenerated. Proof. intros A B. apply hinhfun2. intros Ha Hb. exists (concatenate (pr1 Ha) (pr1 Hb)). split. + simpl ; intros m. unfold concatenate'. set (Hm := (weqfromcoprodofstn_invmap (length (pr1 Ha)) (length (pr1 Hb))) m). change ((weqfromcoprodofstn_invmap (length (pr1 Ha)) (length (pr1 Hb))) m) with Hm. induction Hm as [Hm | Hm]. * rewrite coprod_rect_compute_1. apply (pr1 (pr2 Ha)). * rewrite coprod_rect_compute_2. apply (pr1 (pr2 Hb)). + intros x Hx. simpl in Hx. unfold concatenate' in Hx. split. * apply (pr2 (pr2 Ha)). intros m. specialize (Hx (weqfromcoprodofstn_map _ _ (ii1 m))). rewrite (weqfromcoprodofstn_eq1 _ _), coprod_rect_compute_1 in Hx. exact Hx. * apply (pr2 (pr2 Hb)). intros m. specialize (Hx (weqfromcoprodofstn_map _ _ (ii2 m))). rewrite (weqfromcoprodofstn_eq1 _ _), coprod_rect_compute_2 in Hx. exact Hx. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtergenerated_and | 38,484 |
Lemma filtergenerated_notempty : isfilter_notempty filtergenerated. Proof. intros A. apply hinhuniv. intros L'. generalize (Hl _ (pr1 (pr2 L'))). apply hinhfun. intros x. exists (pr1 x). apply (pr2 (pr2 L')). exact (pr2 x). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | filtergenerated_notempty | 38,485 |
Definition PreFilterGenerated {X : UU} (L : (X β hProp) β hProp) : PreFilter X. Proof. simple refine (make_PreFilter _ _ _ _). - apply (filtergenerated L). - apply filtergenerated_imply. - apply filtergenerated_htrue. - apply filtergenerated_and. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterGenerated | 38,486 |
Definition FilterGenerated {X : UU} (L : (X β hProp) β hProp) (Hl : β L' : Sequence (X β hProp), (β m : stn (length L'), L (L' m)) β β x : X, finite_intersection L' x) : Filter X. Proof. exists (PreFilterGenerated L). split. - apply (pr2 (PreFilterGenerated L)). - apply filtergenerated_notempty. exact Hl. Defined. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterGenerated | 38,487 |
Lemma PreFilterGenerated_correct {X : UU} : β (L : (X β hProp) β hProp), (β A : X β hProp, L A β (PreFilterGenerated L) A) Γ (β F : PreFilter X, (β A : X β hProp, L A β F A) β filter_le F (PreFilterGenerated L)). Proof. intros L. split. - intros A La. apply hinhpr. exists (singletonSequence A). split. + intros. assumption. + intros x Hx. apply (Hx (O,,paths_refl _)). - intros F Hf A. apply hinhuniv. intros Ha. refine (filter_imply _ _ _ _ _). + apply (pr2 (pr2 Ha)). + apply filter_finite_intersection. intros m. apply Hf. apply (pr1 (pr2 Ha)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | PreFilterGenerated_correct | 38,488 |
Lemma FilterGenerated_correct {X : UU} : β (L : (X β hProp) β hProp) (Hl : β L' : Sequence (X β hProp), (β m, L (L' m)) β (β x : X, finite_intersection L' x)), (β A : X β hProp, L A β (FilterGenerated L Hl) A) Γ (β F : Filter X, (β A : X β hProp, L A β F A) β filter_le F (FilterGenerated L Hl)). Proof. intros L Hl. split. - intros A La. apply hinhpr. exists (singletonSequence A). split. + intros; assumption. + intros x Hx. apply (Hx (O,,paths_refl _)). - intros F Hf A. apply hinhuniv. intros Ha. refine (filter_imply _ _ _ _ _). + apply (pr2 (pr2 Ha)). + apply filter_finite_intersection. intros m. apply Hf. apply (pr1 (pr2 Ha)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterGenerated_correct | 38,489 |
Lemma FilterGenerated_inv {X : UU} : β (L : (X β hProp) β hProp) (F : Filter X), (β A : X β hProp, L A β F A) β β (L' : Sequence (X β hProp)), (β m, L (L' m)) β (β x : X, finite_intersection L' x). Proof. intros L F Hf L' Hl'. apply (filter_notempty F). apply filter_finite_intersection. intros m. apply Hf, Hl'. Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | FilterGenerated_inv | 38,490 |
Lemma ex_filter_le {X : UU} : β (F : Filter X) (A : X β hProp), (β G : Filter X, filter_le G F Γ G A) <-> (β B : X β hProp, F B β (β x : X, A x β§ B x)). Proof. intros F A. split. - intros G B Fb. apply (filter_notempty (pr1 G)). apply filter_and. + apply (pr2 (pr2 G)). + now apply (pr1 (pr2 G)). - intros H. simple refine (tpair _ _ _). + simple refine (FilterGenerated _ _). * intros B. apply (F B β¨ B = A). * intros L Hl. assert (B : β B : X β hProp, F B Γ (β x, (A x β§ B x β A x β§ finite_intersection L x))). { revert L Hl. apply (Sequence_rect (P := Ξ» L : Sequence (X β hProp), (β m : stn (length L), (Ξ» B : X β hProp, F B β¨ B = A) (L m)) β β B : X β hProp, F B Γ (β x : X, A x β§ B x β A x β§ finite_intersection L x))). - intros Hl. apply hinhpr. rewrite finite_intersection_htrue. exists (Ξ» _, htrue). split. + exact (filter_htrue F). + intros; assumption. - intros L B IHl Hl. rewrite finite_intersection_append. simple refine (hinhuniv _ _). 3: apply IHl. + intros C. generalize (Hl lastelement) ; simpl. rewrite append_vec_compute_2. apply hinhfun. apply sumofmaps ; [intros Fl | intros ->]. * refine (tpair _ _ _). split. ** apply (filter_and F). *** apply (pr1 (pr2 C)). *** apply Fl. ** intros x H0 ; repeat split. *** exact (pr1 H0). *** exact (pr2 (pr2 H0)). *** simple refine (pr2 (pr2 (pr2 C) x _)). split. **** exact (pr1 H0). **** exact (pr1 (pr2 H0)). * exists (pr1 C) ; split. ** exact (pr1 (pr2 C)). ** intros x H0 ; repeat split. *** exact (pr1 H0). *** exact (pr1 H0). *** simple refine (pr2 (pr2 (pr2 C) x _)). exact H0. + intros. generalize (Hl (dni_lastelement m)) ; simpl. rewrite <- replace_dni_last. now rewrite append_vec_compute_1. } revert B. apply hinhuniv. intros B. generalize (H (pr1 B) (pr1 (pr2 B))). apply hinhfun. intros x. exists (pr1 x). simple refine (pr2 (pr2 (pr2 B) (pr1 x) _)). exact (pr2 x). + split. * intros B Fb. apply hinhpr. exists (singletonSequence B). split. ** intros m. apply hinhpr. now left. ** intros x Hx. apply (Hx (O ,, paths_refl _)). * apply hinhpr. exists (singletonSequence A). split. ** intros m. apply hinhpr. now right. ** intros x Hx. apply (Hx (O ,, paths_refl _)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | ex_filter_le | 38,491 |
Definition isbase_and := β A B : X β hProp, base A β base B β β C : X β hProp, base C Γ (β x, C x β A x β§ B x). | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isbase_and | 38,492 |
Definition isbase_notempty := β A : X β hProp, base A. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isbase_notempty | 38,493 |
Definition isbase_notfalse := β A, base A β β x, A x. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isbase_notfalse | 38,494 |
Definition isBaseOfPreFilter := isbase_and Γ isbase_notempty. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isBaseOfPreFilter | 38,495 |
Definition isBaseOfFilter := isbase_and Γ isbase_notempty Γ isbase_notfalse. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | isBaseOfFilter | 38,496 |
Definition BaseOfPreFilter (X : UU) := β (base : (X β hProp) β hProp), isBaseOfPreFilter base. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | BaseOfPreFilter | 38,497 |
Definition pr1BaseOfPreFilter {X : UU} : BaseOfPreFilter X β ((X β hProp) β hProp) := pr1. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | pr1BaseOfPreFilter | 38,498 |
Definition BaseOfFilter (X : UU) := β (base : (X β hProp) β hProp), isBaseOfFilter base. | Definition | Topology | Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. | Topology\Filters.v | BaseOfFilter | 38,499 |