fact
stringlengths 13
15.5k
| type
stringclasses 9
values | library
stringclasses 15
values | imports
stringlengths 14
7.64k
β | filename
stringlengths 12
97
| symbolic_name
stringlengths 1
78
| index_level
int64 0
38.7k
|
|---|---|---|---|---|---|---|
Lemma topologydirprod_imply : β x : U Γ V, isfilter_imply (topologydirprod x). Proof. intros x A B H. apply hinhfun. intros AB. exists (pr1 AB), (pr1 (pr2 AB)) ; split ; [ | split]. - exact (pr1 (pr2 (pr2 AB))). - exact (pr1 (pr2 (pr2 (pr2 AB)))). - intros x' y' Hx' Hy'. now apply H, (pr2 (pr2 (pr2 (pr2 AB)))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologydirprod_imply | 38,600 |
Lemma topologydirprod_htrue : β x : U Γ V, isfilter_htrue (topologydirprod x). Proof. intros z. apply hinhpr. exists (Ξ» _, htrue), (Ξ» _, htrue). repeat split. - apply isOpen_htrue. - apply isOpen_htrue. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologydirprod_htrue | 38,601 |
Lemma topologydirprod_and : β x : U Γ V, isfilter_and (topologydirprod x). Proof. intros z A B. apply hinhfun2. intros A' B'. exists (Ξ» x, pr1 A' x β§ pr1 B' x), (Ξ» y, pr1 (pr2 A') y β§ pr1 (pr2 B') y). repeat split. - apply (pr1 (pr1 (pr2 (pr2 A')))). - apply (pr1 (pr1 (pr2 (pr2 B')))). - apply isOpen_and. + apply (pr2 (pr1 (pr2 (pr2 A')))). + apply (pr2 (pr1 (pr2 (pr2 B')))). - apply (pr1 (pr1 (pr2 (pr2 (pr2 A'))))). - apply (pr1 (pr1 (pr2 (pr2 (pr2 B'))))). - apply isOpen_and. + apply (pr2 (pr1 (pr2 (pr2 (pr2 A'))))). + apply (pr2 (pr1 (pr2 (pr2 (pr2 B'))))). - apply (pr2 (pr2 (pr2 (pr2 A')))). + apply (pr1 X). + apply (pr1 X0). - apply (pr2 (pr2 (pr2 (pr2 B')))). + apply (pr2 X). + apply (pr2 X0). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologydirprod_and | 38,602 |
Lemma topologydirprod_point : β (x : U Γ V) (P : U Γ V β hProp), topologydirprod x P β P x. Proof. intros xy A. apply hinhuniv. intros A'. apply (pr2 (pr2 (pr2 (pr2 A')))). - exact (pr1 (pr1 (pr2 (pr2 A')))). - exact (pr1 (pr1 (pr2 (pr2 (pr2 A'))))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologydirprod_point | 38,603 |
Lemma topologydirprod_neighborhood : β (x : U Γ V) (P : U Γ V β hProp), topologydirprod x P β β Q : U Γ V β hProp, topologydirprod x Q Γ (β y : U Γ V, Q y β topologydirprod y P). Proof. intros xy P. apply hinhfun. intros A'. exists (Ξ» z, pr1 A' (pr1 z) β§ pr1 (pr2 A') (pr2 z)). split. - apply hinhpr. exists (pr1 A'), (pr1 (pr2 A')). split. + exact (pr1 (pr2 (pr2 A'))). + split. * exact (pr1 (pr2 (pr2 (pr2 A')))). * intros x' y' Ax' Ay'. now split. - intros z Az. apply hinhpr. exists (pr1 A'), (pr1 (pr2 A')). repeat split. + exact (pr1 Az). + exact (pr2 (pr1 (pr2 (pr2 A')))). + exact (pr2 Az). + exact (pr2 (pr1 (pr2 (pr2 (pr2 A'))))). + exact (pr2 (pr2 (pr2 (pr2 A')))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologydirprod_neighborhood | 38,604 |
Definition TopologyDirprod (U V : TopologicalSpace) : TopologicalSpace. Proof. simple refine (TopologyFromNeighborhood _ _). - apply (U Γ V)%set. - apply topologydirprod. - repeat split. + apply topologydirprod_imply. + apply topologydirprod_htrue. + apply topologydirprod_and. + apply topologydirprod_point. + apply topologydirprod_neighborhood. Defined. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | TopologyDirprod | 38,605 |
Definition locally2d {T S : TopologicalSpace} (x : T) (y : S) : Filter (T Γ S) := FilterDirprod (locally x) (locally y). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally2d | 38,606 |
Lemma locally2d_correct {T S : TopologicalSpace} (x : T) (y : S) : β P : T Γ S β hProp, locally2d x y P <-> locally (T := TopologyDirprod T S) (x,,y) P. Proof. intros P. split ; apply hinhuniv. - intros A. apply TopologyFromNeighborhood_correct. generalize (pr1 (pr2 (pr2 A))) (pr1 (pr2 (pr2 (pr2 A)))). apply hinhfun2. intros Ox Oy. exists (pr1 Ox), (pr1 Oy). repeat split. + exact (pr1 (pr2 Ox)). + exact (pr2 (pr1 Ox)). + exact (pr1 (pr2 Oy)). + exact (pr2 (pr1 Oy)). + intros x' y' Hx' Hy'. apply (pr2 (pr2 (pr2 (pr2 A)))). * now apply (pr2 (pr2 Ox)). * now apply (pr2 (pr2 Oy)). - intros O. generalize (pr2 (pr1 O) _ (pr1 (pr2 O))). apply hinhfun. intros A. exists (pr1 A), (pr1 (pr2 A)). repeat split. + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 A)))). * exact (pr1 (pr1 (pr2 (pr2 A)))). + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 (pr2 A))))). * exact (pr1 (pr1 (pr2 (pr2 (pr2 A))))). + intros x' y' Ax' Ay'. apply (pr2 (pr2 O)). now apply (pr2 (pr2 (pr2 (pr2 A)))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally2d_correct | 38,607 |
Definition topologysubtype := Ξ» (x : β x : T, dom x) (A : (β x0 : T, dom x0) β hProp), β B : T β hProp, (B (pr1 x) Γ isOpen B) Γ (β y : β x0 : T, dom x0, B (pr1 y) β A y). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype | 38,608 |
Lemma topologysubtype_imply : β x : β x : T, dom x, isfilter_imply (topologysubtype x). Proof. intros x A B H. apply hinhfun. intros A'. exists (pr1 A'). split. - exact (pr1 (pr2 A')). - intros y Hy. now apply H, (pr2 (pr2 A')). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype_imply | 38,609 |
Lemma topologysubtype_htrue : β x : β x : T, dom x, isfilter_htrue (topologysubtype x). Proof. intros x. apply hinhpr. exists (Ξ» _, htrue). repeat split. now apply isOpen_htrue. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype_htrue | 38,610 |
Lemma topologysubtype_and : β x : β x : T, dom x, isfilter_and (topologysubtype x). Proof. intros x A B. apply hinhfun2. intros A' B'. exists (Ξ» x, pr1 A' x β§ pr1 B' x). repeat split. - exact (pr1 (pr1 (pr2 A'))). - exact (pr1 (pr1 (pr2 B'))). - apply isOpen_and. + exact (pr2 (pr1 (pr2 A'))). + exact (pr2 (pr1 (pr2 B'))). - apply (pr2 (pr2 A')), (pr1 X). - apply (pr2 (pr2 B')), (pr2 X). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype_and | 38,611 |
Lemma topologysubtype_point : β (x : β x : T, dom x) (P : (β x0 : T, dom x0) β hProp), topologysubtype x P β P x. Proof. intros x A. apply hinhuniv. intros B. apply (pr2 (pr2 B)), (pr1 (pr1 (pr2 B))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype_point | 38,612 |
Lemma topologysubtype_neighborhood : β (x : β x : T, dom x) (P : (β x0 : T, dom x0) β hProp), topologysubtype x P β β Q : (β x0 : T, dom x0) β hProp, topologysubtype x Q Γ (β y : β x0 : T, dom x0, Q y β topologysubtype y P). Proof. intros x A. apply hinhfun. intros B. exists (Ξ» y : β x : T, dom x, pr1 B (pr1 y)). split. - apply hinhpr. exists (pr1 B). split. + exact (pr1 (pr2 B)). + intros. assumption. - intros y By. apply hinhpr. exists (pr1 B). repeat split. + exact By. + exact (pr2 (pr1 (pr2 B))). + exact (pr2 (pr2 B)). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | topologysubtype_neighborhood | 38,613 |
Definition TopologySubtype {T : TopologicalSpace} (dom : T β hProp) : TopologicalSpace. Proof. simple refine (TopologyFromNeighborhood _ _). - exact (carrier_subset dom). - apply topologysubtype. - repeat split. + apply topologysubtype_imply. + apply topologysubtype_htrue. + apply topologysubtype_and. + apply topologysubtype_point. + apply topologysubtype_neighborhood. Defined. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | TopologySubtype | 38,614 |
Lemma locally_base_imply : isfilter_imply (neighborhood' x base). Proof. intros A B H Ha. apply (pr2 (neighborhood_equiv _ _ _)). eapply neighborhood_imply. - exact H. - eapply neighborhood_equiv. exact Ha. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally_base_imply | 38,615 |
Lemma locally_base_htrue : isfilter_htrue (neighborhood' x base). Proof. apply (pr2 (neighborhood_equiv _ _ _)). apply (pr2 (neighborhood_isOpen _)). - apply isOpen_htrue. - apply tt. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally_base_htrue | 38,616 |
Lemma locally_base_and : isfilter_and (neighborhood' x base). Proof. intros A B Ha Hb. apply (pr2 (neighborhood_equiv _ _ _)). eapply neighborhood_and. - eapply neighborhood_equiv, Ha. - eapply neighborhood_equiv, Hb. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally_base_and | 38,617 |
Definition locally_base {T : TopologicalSpace} (x : T) (base : base_of_neighborhood x) : Filter T. Proof. simple refine (make_Filter _ _ _ _ _). - apply (neighborhood' x base). - apply locally_base_imply. - apply locally_base_htrue. - apply locally_base_and. - intros A Ha. apply neighborhood_equiv in Ha. apply neighborhood_point in Ha. apply hinhpr. now exists x. Defined. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | locally_base | 38,618 |
Definition is_filter_lim {T : TopologicalSpace} (F : Filter T) (x : T) := filter_le F (locally x). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_filter_lim | 38,619 |
Definition ex_filter_lim {T : TopologicalSpace} (F : Filter T) := β (x : T), is_filter_lim F x. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_filter_lim | 38,620 |
Definition is_filter_lim_base {T : TopologicalSpace} (F : Filter T) (x : T) base := filter_le F (locally_base x base). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_filter_lim_base | 38,621 |
Definition ex_filter_lim_base {T : TopologicalSpace} (F : Filter T) := β (x : T) base, is_filter_lim_base F x base. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_filter_lim_base | 38,622 |
Lemma is_filter_lim_base_correct {T : TopologicalSpace} (F : Filter T) (x : T) base : is_filter_lim_base F x base <-> is_filter_lim F x. Proof. split. - intros Hx P HP. apply (pr2 (neighborhood_equiv _ base _)) in HP. apply Hx. exact HP. - intros Hx P HP. apply neighborhood_equiv in HP. apply Hx. exact HP. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_filter_lim_base_correct | 38,623 |
Lemma ex_filter_lim_base_correct {T : TopologicalSpace} (F : Filter T) : ex_filter_lim_base F <-> ex_filter_lim F. Proof. split. - apply hinhfun. intros x. exists (pr1 x). eapply is_filter_lim_base_correct. exact (pr2 (pr2 x)). - apply hinhfun. intros x. exists (pr1 x), (base_of_neighborhood_default (pr1 x)). apply (pr2 (is_filter_lim_base_correct _ _ _)). exact (pr2 x). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_filter_lim_base_correct | 38,624 |
Definition is_lim {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) (x : T) := filterlim f F (locally x). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_lim | 38,625 |
Definition ex_lim {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) := β (x : T), is_lim f F x. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_lim | 38,626 |
Definition is_lim_base {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) (x : T) base := filterlim f F (locally_base x base). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_lim_base | 38,627 |
Definition ex_lim_base {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) := β (x : T) base, is_lim_base f F x base. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_lim_base | 38,628 |
Lemma is_lim_base_correct {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) (x : T) base : is_lim_base f F x base <-> is_lim f F x. Proof. split. - intros Hx P HP. apply Hx, (pr2 (neighborhood_equiv _ _ _)). exact HP. - intros Hx P HP. eapply Hx, neighborhood_equiv. exact HP. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | is_lim_base_correct | 38,629 |
Lemma ex_lim_base_correct {X : UU} {T : TopologicalSpace} (f : X β T) (F : Filter X) : ex_lim_base f F <-> ex_lim f F. Proof. split. - apply hinhfun. intros x. exists (pr1 x). eapply is_lim_base_correct. exact (pr2 (pr2 x)). - apply hinhfun. intros x. exists (pr1 x), (base_of_neighborhood_default (pr1 x)). apply (pr2 (is_lim_base_correct _ _ _ _)). exact (pr2 x). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | ex_lim_base_correct | 38,630 |
Definition continuous_at {U V : TopologicalSpace} (f : U β V) (x : U) := is_lim f (locally x) (f x). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_at | 38,631 |
Definition continuous_on {U V : TopologicalSpace} (dom : U β hProp) (f : β (x : U), dom x β V) := β (x : U) (Hx : dom x), β H, is_lim (Ξ» y : (β x : U, dom x), f (pr1 y) (pr2 y)) (FilterSubtype (locally x) dom H) (f x Hx). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_on | 38,632 |
Definition continuous {U V : TopologicalSpace} (f : U β V) := β x : U, continuous_at f x. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous | 38,633 |
Lemma isaprop_continuous (x y : TopologicalSpace) (f : x β y) : isaprop (continuous (Ξ» x0 : x, f x0)). Proof. do 3 (apply impred_isaprop; intro). apply propproperty. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isaprop_continuous | 38,634 |
Definition continuous_base_at {U V : TopologicalSpace} (f : U β V) (x : U) base_x base_fx := is_lim_base f (locally_base x base_x) (f x) base_fx. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_base_at | 38,635 |
Definition continuous2d_at {U V W : TopologicalSpace} (f : U β V β W) (x : U) (y : V) := is_lim (Ξ» z : U Γ V, f (pr1 z) (pr2 z)) (FilterDirprod (locally x) (locally y)) (f x y). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous2d_at | 38,636 |
Definition continuous2d {U V W : TopologicalSpace} (f : U β V β W) := β (x : U) (y : V), continuous2d_at f x y. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous2d | 38,637 |
Definition continuous2d_base_at {U V W : TopologicalSpace} (f : U β V β W) (x : U) (y : V) base_x base_y base_fxy := is_lim_base (Ξ» z : U Γ V, f (pr1 z) (pr2 z)) (FilterDirprod (locally_base x base_x) (locally_base y base_y)) (f x y) base_fxy. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous2d_base_at | 38,638 |
Lemma continuous_comp {X : UU} {U V : TopologicalSpace} (f : X β U) (g : U β V) (F : Filter X) (l : U) : is_lim f F l β continuous_at g l β is_lim (funcomp f g) F (g l). Proof. apply filterlim_comp. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_comp | 38,639 |
Lemma continuous_funcomp {X Y Z : TopologicalSpace} (f : X β Y) (g : Y β Z) : continuous f β continuous g β continuous (funcomp f g). Proof. intros Hf Hg x. refine (continuous_comp _ _ _ _ _ _). - apply Hf. - apply Hg. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_funcomp | 38,640 |
Lemma continuous2d_comp {X : UU} {U V W : TopologicalSpace} (f : X β U) (g : X β V) (h : U β V β W) (F : Filter X) (lf : U) (lg : V) : is_lim f F lf β is_lim g F lg β continuous2d_at h lf lg β is_lim (Ξ» x, h (f x) (g x)) F (h lf lg). Proof. intros Hf Hg. apply (filterlim_comp (Ξ» x, (f x ,, g x))). intros P. apply hinhuniv. intros Hp. generalize (filter_and F _ _ (Hf _ (pr1 (pr2 (pr2 Hp)))) (Hg _ (pr1 (pr2 (pr2 (pr2 Hp)))))). apply (filter_imply F). intros x Hx. apply (pr2 (pr2 (pr2 (pr2 Hp)))). - exact (pr1 Hx). - exact (pr2 Hx). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous2d_comp | 38,641 |
Lemma continuous_tpair {U V : TopologicalSpace} : continuous2d (W := TopologyDirprod U V) (Ξ» (x : U) (y : V), (x,,y)). Proof. intros x y P. apply hinhuniv. intros O. simple refine (filter_imply _ _ _ _ _). - exact (pr1 O). - exact (pr2 (pr2 O)). - generalize (pr2 (pr1 O) _ (pr1 (pr2 O))). apply hinhfun. intros Ho. exists (pr1 Ho), (pr1 (pr2 Ho)). repeat split. + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 Ho)))). * exact (pr1 (pr1 (pr2 (pr2 Ho)))). + apply (pr2 (neighborhood_isOpen _)). * exact (pr2 (pr1 (pr2 (pr2 (pr2 Ho))))). * exact (pr1 (pr1 (pr2 (pr2 (pr2 Ho))))). + exact (pr2 (pr2 (pr2 (pr2 Ho)))). Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_tpair | 38,642 |
Lemma continuous_pr1 {U V : TopologicalSpace} : continuous (U := TopologyDirprod U V) (Ξ» (xy : U Γ V), pr1 xy). Proof. intros xy P. apply hinhuniv. intros O. simple refine (filter_imply _ _ _ _ _). - exact (pr1 (pr1 O)). - exact (pr2 (pr2 O)). - apply hinhpr. use tpair. + use tpair. * apply (Ξ» xy : U Γ V, pr1 O (pr1 xy)). * intros xy' Oxy. apply hinhpr. exists (pr1 O), (Ξ» _, htrue). repeat split. ** exact Oxy. ** exact (pr2 (pr1 O)). ** exact isOpen_htrue. ** intros. assumption. + repeat split. * exact (pr1 (pr2 O)). * intros. assumption. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_pr1 | 38,643 |
Lemma continuous_pr2 {U V : TopologicalSpace} : continuous (U := TopologyDirprod U V) (Ξ» (xy : U Γ V), pr2 xy). Proof. intros xy P. apply hinhuniv. intros O. simple refine (filter_imply _ _ _ _ _). - exact (pr1 (pr1 O)). - exact (pr2 (pr2 O)). - apply hinhpr. use tpair. + use tpair. * apply (Ξ» xy : U Γ V, pr1 O (pr2 xy)). * intros xy' Oxy. apply hinhpr. exists (Ξ» _, htrue), (pr1 O). repeat split. ** exact isOpen_htrue. ** exact Oxy. ** exact (pr2 (pr1 O)). ** intros. assumption. + repeat split. * exact (pr1 (pr2 O)). * intros. assumption. Qed. | Lemma | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | continuous_pr2 | 38,644 |
Definition isTopological_monoid (X : monoid) (is : isTopologicalSpace X) := continuous2d (U := (pr11 X) ,, is) (V := (pr11 X) ,, is) (W := (pr11 X) ,, is) BinaryOperations. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_monoid | 38,645 |
Definition Topological_monoid := β (X : monoid) (is : isTopologicalSpace X), isTopological_monoid X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_monoid | 38,646 |
Definition isTopological_gr (X : gr) (is : isTopologicalSpace X) := isTopological_monoid X is Γ continuous (U := (pr11 X) ,, is) (V := (pr11 X) ,, is) (grinv X). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_gr | 38,647 |
Definition Topological_gr := β (X : gr) is, isTopological_gr X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_gr | 38,648 |
Definition isTopological_rig (X : rig) (is : isTopologicalSpace X) := isTopological_monoid (rigaddabmonoid X) is Γ isTopological_monoid (rigmultmonoid X) is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_rig | 38,649 |
Definition Topological_rig := β (X : rig) is, isTopological_rig X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_rig | 38,650 |
Definition isTopological_ring (X : ring) (is : isTopologicalSpace X) := isTopological_gr (ringaddabgr X) is Γ isTopological_monoid (rigmultmonoid X) is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_ring | 38,651 |
Definition Topological_ring := β (X : ring) is, isTopological_ring X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_ring | 38,652 |
Definition isTopological_DivRig (X : DivRig) (is : isTopologicalSpace X) := isTopological_rig (pr1 X) is Γ continuous_on (U := (pr111 X) ,, is) (V := (pr111 X) ,, is) (Ξ» x : X, make_hProp (x != 0%dr) (isapropneg _)) (Ξ» x Hx, invDivRig (x,,Hx)). | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_DivRig | 38,653 |
Definition Topological_DivRig := β (X : DivRig) is, isTopological_DivRig X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_DivRig | 38,654 |
Definition isTopological_fld (X : fld) (is : isTopologicalSpace X) := isTopological_ring (pr1 X) is Γ continuous_on (U := (pr111 X) ,, is) (V := (pr111 X) ,, is) (Ξ» x : X, make_hProp (x != 0%ring) (isapropneg _)) fldmultinv. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_fld | 38,655 |
Definition Topological_fld := β (X : fld) is, isTopological_fld X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_fld | 38,656 |
Definition isTopological_ConstructiveDivisionRig (X : ConstructiveDivisionRig) (is : isTopologicalSpace X) := isTopological_rig X is Γ continuous_on (U := (pr111 X) ,, is) (V := (pr111 X) ,, is) (Ξ» x : X, (x β 0)%CDR) CDRinv. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_ConstructiveDivisionRig | 38,657 |
Definition Topological_ConstructiveDivisionRig := β (X : ConstructiveDivisionRig) is, isTopological_ConstructiveDivisionRig X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_ConstructiveDivisionRig | 38,658 |
Definition isTopological_ConstructiveField (X : ConstructiveField) (is : isTopologicalSpace X) := isTopological_ring X is Γ continuous_on (U := (pr111 X) ,, is) (V := (pr111 X) ,, is) (Ξ» x : X, (x β 0)%CF) CFinv. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | isTopological_ConstructiveField | 38,659 |
Definition Topological_ConstructiveField := β (X : ConstructiveField) is, isTopological_ConstructiveField X is. | Definition | Topology | Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). | Topology\Topology.v | Topological_ConstructiveField | 38,660 |