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Coq-UniMath

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Lemma topologydirprod_imply : ∏ 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 : ∏ 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 : ∏ x : U Γ— V, isfilter_and (topologydirprod x). Proof. intros z A B. apply hinhfun2. intros A' B'. exists (Ξ» x, pr1 A' x ∧ pr1 B' x), (Ξ» y, pr1 (pr2 A') y ∧ 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 : ∏ (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 : ∏ (x : U Γ— V) (P : U Γ— V β†’ hProp), topologydirprod x P β†’ βˆƒ Q : U Γ— V β†’ hProp, topologydirprod x Q Γ— (∏ 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) ∧ 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) : ∏ 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 : βˆ‘ x : T, dom x) (A : (βˆ‘ x0 : T, dom x0) β†’ hProp), βˆƒ B : T β†’ hProp, (B (pr1 x) Γ— isOpen B) Γ— (∏ y : βˆ‘ 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 : ∏ x : βˆ‘ 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 : ∏ x : βˆ‘ 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 : ∏ x : βˆ‘ x : T, dom x, isfilter_and (topologysubtype x). Proof. intros x A B. apply hinhfun2. intros A' B'. exists (Ξ» x, pr1 A' x ∧ 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 : ∏ (x : βˆ‘ x : T, dom x) (P : (βˆ‘ 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 : ∏ (x : βˆ‘ x : T, dom x) (P : (βˆ‘ x0 : T, dom x0) β†’ hProp), topologysubtype x P β†’ βˆƒ Q : (βˆ‘ x0 : T, dom x0) β†’ hProp, topologysubtype x Q Γ— (∏ y : βˆ‘ x0 : T, dom x0, Q y β†’ topologysubtype y P). Proof. intros x A. apply hinhfun. intros B. exists (Ξ» y : βˆ‘ 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) := βˆƒ (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) := βˆƒ (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) := βˆƒ (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) := βˆƒ (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 : ∏ (x : U), dom x β†’ V) := ∏ (x : U) (Hx : dom x), βˆƒ H, is_lim (Ξ» y : (βˆ‘ 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) := ∏ 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) := ∏ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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 := βˆ‘ (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