fact stringlengths 9 12.1k | type stringclasses 25
values | library stringclasses 6
values | imports listlengths 0 15 | filename stringclasses 105
values | symbolic_name stringlengths 1 50 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
closure_limit_point := __deprecated__closure_limit_point (only parsing). | Notation | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_limit_point | |
closed_comp {T U : topologicalType} (f : T -> U) (D : set U) : {in ~` f @^-1` D, continuous f} -> closed D -> closed (f @^-1` D). Proof. rewrite !closedE=> f_continuous D_cl x /= xDf. apply: D_cl; apply: contra_not xDf => fxD. have NDfx : ~ D (f x). by move: fxD; rewrite -nbhs_nearE nbhsE => - [A [? ?]]; exact. by appl... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closed_comp | |
closed_cvg {T} {V : topologicalType} {F} {FF : ProperFilter F} (u_ : T -> V) (A : V -> Prop) : (* BUG: elim does not see this as an elimination principle if A : set V *) closed A -> (\forall n \near F, A (u_ n)) -> forall l, u_ @ F --> l -> A l. Proof. move=> + FAu_ l u_Fl; apply => B /u_Fl /=; rewrite nbhs_filterE. by... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closed_cvg | |
continuous_closedP (S T : topologicalType) (f : S -> T) : continuous f <-> forall A, closed A -> closed (f @^-1` A). Proof. rewrite continuousP; split=> ctsf ? ?. by rewrite -openC preimage_setC; apply: ctsf; rewrite openC. by rewrite -closedC preimage_setC; apply: ctsf; rewrite closedC. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | continuous_closedP | |
closedU (T : topologicalType) (D E : set T) : closed D -> closed E -> closed (D `|` E). Proof. by rewrite -?openC setCU; exact: openI. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closedU | |
closed_bigsetU (T : topologicalType) (I : eqType) (s : seq I) (F : I -> set T) : (forall x, x \in s -> closed (F x)) -> closed (\big[setU/set0]_(x <- s) F x). Proof. move=> scF; rewrite big_seq. by elim/big_ind : _ => //; [exact: closed0|exact: closedU]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closed_bigsetU | |
closed_bigcup (T : topologicalType) (I : choiceType) (A : set I) (F : I -> set T) : finite_set A -> (forall i, A i -> closed (F i)) -> closed (\bigcup_(i in A) F i). Proof. move=> finA cF; rewrite -bigsetU_fset_set//; apply: closed_bigsetU => i. by rewrite in_fset_set// inE; exact: cF. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closed_bigcup | |
closure_subset A B : A `<=` B -> closure A `<=` closure B. Proof. by move=> ? ? CAx ?; move/CAx; exact/subsetI_neq0. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_subset | |
closureE A : closure A = smallest closed A. Proof. rewrite eqEsubset; split=> [x ? B [cB AB]|]; first exact/cB/(closure_subset AB). exact: (smallest_sub (@closed_closure _ _) (@subset_closure _ _)). Qed. (* TODO: the LHS and RHS of the equality should be swapped *) | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureE | |
closure_id E : closed E <-> E = closure E. Proof. split=> [?|->]; last exact: closed_closure. rewrite eqEsubset; split => //; exact: subset_closure. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_id | |
regopen (A : set T) := (closure A)° = A. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | regopen | |
regclosed (A : set T) := closure (A°) = A. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | regclosed | |
interiorC A : (~` A)° = ~` closure A. Proof. rewrite eqEsubset; split=> x; rewrite /closure /interior nbhsE /= -existsNE. case=> U ? /disjoints_subset UA; exists U; rewrite not_implyE. split; first exact/open_nbhs_nbhs. by rewrite setIC UA; apply/set0P; rewrite eqxx. case=> X; rewrite not_implyE nbhsE=> -[] -[] U xU UX... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interiorC | |
closure_setC A : closure (~` A) = ~` A°. Proof. by apply: setC_inj; rewrite -interiorC !setCK. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_setC | |
interior_id A : open A <-> interior A = A. Proof. by rewrite -(setCK A) openC interiorC closure_id; split => [ <- | /setC_inj->]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interior_id | |
closureT : closure [set: T] = [set: T]. Proof. exact/esym/closure_id/closedT. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureT | |
closure0 : closure (@set0 T) = set0. Proof. exact/esym/closure_id/closed0. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure0 | |
interiorT : (@setT T)° = setT. Proof. exact/interior_id/openT. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interiorT | |
interior0 : (@set0 T)° = set0. Proof. exact/interior_id/open0. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interior0 | |
closureU A B : closure (A `|` B) = closure A `|` closure B. Proof. by apply: setC_inj; rewrite setCU -!interiorC -interiorI setCU. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureU | |
interiorU A B : A° `|` B° `<=` (A `|` B)°. Proof. by apply: subsetC2; rewrite setCU -!closure_setC setCU; exact: closureI. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interiorU | |
closureEbigcap A : closure A = \bigcap_(x in [set C | closed C /\ A `<=` C]) x. Proof. exact: closureE. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureEbigcap | |
interiorEbigcup A : A° = \bigcup_(x in [set U | open U /\ U `<=` A]) x. Proof. apply: setC_inj; rewrite -closure_setC closureEbigcap setC_bigcup. rewrite -[RHS](bigcap_image _ setC idfun) /=. apply: eq_bigcapl; split => X /=. by rewrite -openC -setCS setCK; exists (~` X)=> //; rewrite setCK. by case=> Y + <-; rewrite c... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interiorEbigcup | |
interior_closed_regopen A : closed A -> regopen A°. Proof. move=> cA; rewrite /regopen eqEsubset; split=> x. rewrite /closure [X in X -> _]/interior nbhsE => -[] U oxU UciA. rewrite /interior nbhsE /=; exists U => //. apply: (subset_trans UciA) => y /= yA. apply: cA => B /yA; apply/subset_nonempty; apply: setSI. exact:... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interior_closed_regopen | |
closure_open_regclosed A : open A -> regclosed (closure A). Proof. rewrite /regclosed -(setCK A) openC => cCA. rewrite closure_setC -[in RHS]interior_closed_regopen//. by rewrite !(closure_setC, interiorC). Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_open_regclosed | |
interior_closure_idem : @idempotent_fun (set T) (interior \o closure). Proof. move=> ?; exact/interior_closed_regopen/closed_closure. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | interior_closure_idem | |
closure_interior_idem : @idempotent_fun (set T) (closure \o interior). Proof. move=> ?; exact/closure_open_regclosed/open_interior. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closure_interior_idem | |
closureC_deprecated (T : topologicalType) (E : set T) : ~` closure E = \bigcup_(x in [set U | open U /\ U `<=` ~` E]) x. Proof. by rewrite -interiorC interiorEbigcup. Qed. #[deprecated(since="mathcomp-analysis 1.7.0", note="use `interiorC` and `interiorEbigcup` instead")] | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureC_deprecated | |
closureC := closureC_deprecated (only parsing). | Notation | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | closureC | |
dense (T : topologicalType) (S : set T) := forall (O : set T), O !=set0 -> open O -> O `&` S !=set0. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | dense | |
denseNE (T : topologicalType) (S : set T) : ~ dense S -> exists O, (exists x, open_nbhs x O) /\ (O `&` S = set0). Proof. rewrite /dense /open_nbhs. move=> /existsNP[X /not_implyP[[x Xx] /not_implyP[ Ox /forallNP A]]]. by exists X; split; [exists x | rewrite -subset0; apply/A]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | denseNE | |
denseI (T : topologicalType) (A B : set T) : open A -> dense A -> dense B -> dense (A `&` B). Proof. move=> oA dA dB C C0 oC. have CA0 : C `&` A !=set0 by exact: dA. suff: (C `&` A) `&` B !=set0 by rewrite setIA. by apply: dB => //; exact: openI. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | denseI | |
dense0 {R : ptopologicalType} : ~ dense (@set0 R). Proof. apply/existsNP; exists setT. apply/not_implyP; split; first exact/set0P/setT0. apply/not_implyP; split; first exact: openT. by rewrite setTI => -[]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | dense0 | |
clopen {T} (A : set T) := open A /\ closed A. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopen | |
clopenI {T} (A B : set T) : clopen A -> clopen B -> clopen (A `&` B). Proof. by case=> ? ? [] ? ?; split; [exact: openI | exact: closedI]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopenI | |
clopenU {T} (A B : set T) : clopen A -> clopen B -> clopen (A `|` B). Proof. by case=> ? ? [] ? ?; split; [exact: openU | exact: closedU]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopenU | |
clopenC {T} (A B : set T) : clopen A -> clopen (~`A). Proof. by case=> ? ?; split;[exact: closed_openC | exact: open_closedC ]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopenC | |
clopen0 {T} : @clopen T set0. Proof. by split; [exact: open0 | exact: closed0]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopen0 | |
clopenT {T} : clopen [set: T]. Proof. by split; [exact: openT | exact: closedT]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopenT | |
clopen_comp {T U : topologicalType} (f : T -> U) (A : set U) : clopen A -> continuous f -> clopen (f @^-1` A). Proof. by case=> ? ?; split; [ exact: open_comp | exact: closed_comp]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | clopen_comp | |
continuousEP {X Y : nbhsType} (f g : continuousType X Y) : f = g <-> f =1 g. Proof. case: f g => [f [[ffun]]] [g [[gfun]]]/=; split=> [[->//]|/funext eqfg]. rewrite eqfg in ffun *; congr {| Continuous.sort := _; Continuous.class := {| Continuous.topology_structure_isContinuous_mixin := {|isContinuous.cts_fun := _|}|}|}... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | continuousEP | |
mkcts {X Y : nbhsType} (f : X -> Y) (f_cts : continuous f) := f. HB.instance Definition _ {X Y : nbhsType} (f: X -> Y) (f_cts : continuous f) := @isContinuous.Build X Y (mkcts f_cts) f_cts. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra finmap all_classical.",
"From mathcomp Require Export filter."
] | theories/topology_theory/topology_structure.v | mkcts | |
nbhs_ {T T'} (ent : set_system (T * T')) (x : T) := filter_from ent (fun A => xsection A x). | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_ | |
nbhs_E {T T'} (ent : set_system (T * T')) x : nbhs_ ent x = filter_from ent (fun A => xsection A x). Proof. by []. Qed. Local Open Scope relation_scope. HB.mixin Record Nbhs_isUniform_mixin M of Nbhs M := { entourage : set_system (M * M); entourage_filter : Filter entourage; entourage_diagonal_subproof : forall A, ento... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_E | |
nbhs_filter (p : M) : ProperFilter (nbhs p). Proof. rewrite nbhsE nbhs_E; apply: filter_from_proper; last first. by move=> A entA; exists p; apply/mem_set; apply: entourage_diagonal entA _ _. apply: filter_from_filter. by exists setT; exact: @filterT entourage_filter. move=> A B entA entB; exists (A `&` B); last by rew... | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_filter | |
nbhs_singleton (p : M) A : nbhs p A -> A p. Proof. rewrite nbhsE nbhs_E => - [B entB sBpA]. by apply/sBpA/mem_set; exact: entourage_diagonal entB _ _. Qed. | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_singleton | |
nbhs_nbhs (p : M) A : nbhs p A -> nbhs p (nbhs^~ A). Proof. rewrite nbhsE nbhs_E => - [B entB sBpA]. have /entourage_split_ex[C entC sC2B] := entB. exists C => // q Cpq; rewrite nbhs_E; exists C => // r Cqr. by apply/sBpA/mem_set/sC2B; exists q; exact/set_mem. Qed. HB.instance Definition _ := Nbhs_isNbhsTopological.Bui... | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_nbhs | |
nbhs_entourageE {M : uniformType} : nbhs_ (@entourage M) = nbhs. Proof. by rewrite -Nbhs_isUniform_mixin.nbhsE_subproof. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_entourageE | |
entourage_sym {X Y : Type} E (x : X) (y : Y) : E (x, y) <-> (E ^-1)%relation (y, x). Proof. by []. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_sym | |
filter_from_entourageE {M : uniformType} x : filter_from (@entourage M) (fun A => xsection A x) = nbhs x. Proof. by rewrite -nbhs_entourageE. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | filter_from_entourageE | |
nbhs_simpl := (nbhs_simpl,@filter_from_entourageE,@nbhs_entourageE). | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_simpl | |
nbhsP {M : uniformType} (x : M) P : nbhs x P <-> nbhs_ entourage x P. Proof. by rewrite nbhs_simpl. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhsP | |
filter_inv {T : Type} (F : set (set (T * T))) : Filter F -> Filter [set V^-1 | V in F]%relation. Proof. move=> FF; split => /=. - by exists [set: T * T] => //; exact: filterT. - by move=> P Q [R FR <-] [S FS <-]; exists (R `&` S) => //; exact: filterI. - move=> P Q PQ [R FR RP]; exists Q^-1%relation => //; first last. ... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | filter_inv | |
entourage_refl (A : set (M * M)) x : entourage A -> A (x, x). Proof. by move=> entA; exact: entourage_diagonal_subproof entA _ _. Qed. Global Instance entourage_filter' : Filter (@entourage M). Proof. exact: entourage_filter. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_refl | |
entourageT : entourage [set: M * M]. Proof. exact: filterT. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourageT | |
entourage_inv (A : set (M * M)) : entourage A -> entourage A^-1. Proof. exact: entourage_inv_subproof. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_inv | |
entourage_split_ex (A : set (M * M)) : entourage A -> exists2 B, entourage B & B \; B `<=` A. Proof. exact: entourage_split_ex_subproof. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_split_ex | |
split_ent (A : set (M * M)) := get (entourage `&` [set B | B \; B `<=` A]). | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | split_ent | |
split_entP (A : set (M * M)) : entourage A -> entourage (split_ent A) /\ split_ent A \; split_ent A `<=` A. Proof. by move/entourage_split_ex/exists2P/getPex. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | split_entP | |
entourage_split_ent (A : set (M * M)) : entourage A -> entourage (split_ent A). Proof. by move=> /split_entP []. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_split_ent | |
subset_split_ent (A : set (M * M)) : entourage A -> split_ent A \; split_ent A `<=` A. Proof. by move=> /split_entP []. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | subset_split_ent | |
entourage_split (z x y : M) A : entourage A -> split_ent A (x, z) -> split_ent A (z, y) -> A (x, y). Proof. by move=> /subset_split_ent sA ? ?; apply: sA; exists z. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_split | |
nbhs_entourage (x : M) A : entourage A -> nbhs x (xsection A x). Proof. by move=> ?; apply/nbhsP; exists A. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | nbhs_entourage | |
cvg_entourageP F (FF : Filter F) (p : M) : F --> p <-> forall A, entourage A -> \forall q \near F, A (p, q). Proof. rewrite -filter_fromP [X in filter_from _ X](_ : _ = @xsection M M ^~ p)//. by rewrite filter_from_entourageE. by apply/funext => E; apply/seteqP; split => [|] ? /xsectionP. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cvg_entourageP | |
cvg_entourage {F} {FF : Filter F} (x : M) : F --> x -> forall A, entourage A -> \forall y \near F, A (x, y). Proof. by move/cvg_entourageP. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cvg_entourage | |
cvg_app_entourageP T (f : T -> M) F (FF : Filter F) p : f @ F --> p <-> forall A, entourage A -> \forall t \near F, A (p, f t). Proof. exact: cvg_entourageP. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cvg_app_entourageP | |
entourage_invI (E : set (M * M)) : entourage E -> entourage (E `&` E^-1). Proof. by move=> ?; apply: filterI; last exact: entourage_inv. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_invI | |
split_ent_subset (E : set (M * M)) : entourage E -> split_ent E `<=` E. Proof. move=> entE; case=> x y splitxy; apply: subset_split_ent => //; exists y => //. by apply: entourage_refl; exact: entourage_split_ent. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | split_ent_subset | |
ent_closure {M : uniformType} (x : M) E : entourage E -> closure (xsection (split_ent E) x) `<=` xsection E x. Proof. pose E' := (split_ent E) `&` (split_ent E)^-1%relation. move=> entE z /(_ (xsection E' z))[]. by rewrite -nbhs_entourageE; exists E' => //; exact: filterI. move=> y; rewrite xsectionI => -[/xsectionP xy... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | ent_closure | |
continuous_withinNx {U V : uniformType} (f : U -> V) x : {for x, continuous f} <-> f @ x^' --> f x. Proof. split=> - cfx P /= fxP. by rewrite !near_simpl; apply: cvg_within; apply: cfx. rewrite !nbhs_nearE !near_map !near_nbhs in fxP *; have /= := cfx P fxP. rewrite !near_simpl near_withinE near_simpl => Pf; near=> y. ... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | continuous_withinNx | |
continuous_injective_withinNx (T U : topologicalType) (f : T -> U) (x : T) : {for x, continuous f} -> (forall y, f y = f x -> y = x) -> f @ x^' --> (f x)^'. Proof. move=> cf fI A; rewrite /nbhs /= /dnbhs !withinE/= => -[V Vfx AV]. exists (f @^-1` V); first exact: cf Vfx. by apply/seteqP; split=> y/=; move/predeqP : AV ... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | continuous_injective_withinNx | |
countable_uniformity (T : uniformType) := exists R : set_system (T * T), [/\ countable R, R `<=` entourage & forall P, entourage P -> exists2 Q, R Q & Q `<=` P]. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | countable_uniformity | |
countable_uniformityP {T : uniformType} : countable_uniformity T <-> exists2 f : nat -> set (T * T), (forall A, entourage A -> exists N, f N `<=` A) & (forall n, entourage (f n)). Proof. split=> [[M []]|[f fsubE entf]]. move=> /pfcard_geP[-> _ /(_ _ (@entourageT _))[]//|/unsquash f eM Msub]. exists f; last by move=> n;... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | countable_uniformityP | |
open_nbhs_entourage (U : uniformType) (x : U) (A : set (U * U)) : entourage A -> open_nbhs x (xsection A x)°. Proof. move=> entA; split; first exact: open_interior. by apply: nbhs_singleton; apply: nbhs_interior; exact: nbhs_entourage. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | open_nbhs_entourage | |
unif_continuous (U V : uniformType) (f : U -> V) := (fun xy => (f xy.1, f xy.2)) @ entourage --> entourage. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | unif_continuous | |
entourage_set (U : uniformType) (A : set ((set U) * (set U))) := exists2 B, entourage B & forall PQ, A PQ -> forall p q, PQ.1 p -> PQ.2 q -> B (p,q). (** Complete uniform spaces *) | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | entourage_set | |
cauchy {T : uniformType} (F : set_system T) := (F, F) --> entourage. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cauchy | |
cvg_cauchy {T : puniformType} (F : set_system T) : Filter F -> [cvg F in T] -> cauchy F. Proof. move=> FF cvF A entA; have /entourage_split_ex [B entB sB2A] := entA. exists (xsection (B^-1%relation) (lim F), xsection B (lim F)). split=> /=; apply: cvF; rewrite /= -nbhs_entourageE; last by exists B. by exists B^-1%relat... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cvg_cauchy | |
complete_ax := cauchy_cvg (only parsing). | Notation | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | complete_ax | |
cauchy_cvgP (F : set_system T) (FF : ProperFilter F) : cauchy F <-> cvg F. Proof. by split=> [/cauchy_cvg|/cvg_cauchy]. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical.",
"From mathcomp Require Import topology_structure."
] | theories/topology_theory/uniform_structure.v | cauchy_cvgP | |
weak_topology {S : Type} {T : Type} (f : S -> T) : Type := S. | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_topology | |
wopen := [set f @^-1` A | A in open]. Local Lemma wopT : wopen [set: W]. Proof. by exists setT => //; apply: openT. Qed. Local Lemma wopI (A B : set W) : wopen A -> wopen B -> wopen (A `&` B). Proof. by move=> [C Cop <-] [D Dop <-]; exists (C `&` D) => //; apply: openI. Qed. Local Lemma wop_bigU (I : Type) (g : I -> se... | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | wopen | |
weak_continuous : continuous (f : W -> T). Proof. by apply/continuousP => A ?; exists A. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_continuous | |
cvg_image (F : set_system S) (s : S) : Filter F -> f @` setT = setT -> F --> (s : W) <-> ([set f @` A | A in F] : set_system _) --> f s. Proof. move=> FF fsurj; split=> [cvFs|cvfFfs]. move=> A /weak_continuous [B [Bop Bs sBAf]]. have /cvFs FB : nbhs (s : W) B by apply: open_nbhs_nbhs. rewrite nbhs_simpl; exists (f @^-1... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | cvg_image | |
S := weak_topology f. | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | S | |
weak_ent : set_system (S * S) := filter_from (@entourage U) (fun V => (map_pair f)@^-1` V). | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent | |
weak_ent_filter : Filter weak_ent. Proof. apply: filter_from_filter; first by exists setT; exact: entourageT. by move=> P Q ??; (exists (P `&` Q); first exact: filterI) => ?. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent_filter | |
weak_ent_refl A : weak_ent A -> diagonal `<=` A. Proof. by move=> [B ? sBA] [x y]/diagonalP ->; apply/sBA; exact: entourage_refl. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent_refl | |
weak_ent_inv A : weak_ent A -> weak_ent A^-1. Proof. move=> [B ? sBA]; exists B^-1; first exact: entourage_inv. by move=> ??; exact/sBA. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent_inv | |
weak_ent_split A : weak_ent A -> exists2 B, weak_ent B & B \; B `<=` A. Proof. move=> [B entB sBA]; have : exists C, entourage C /\ C \; C `<=` B. exact/exists2P/entourage_split_ex. case=> C [entC CsubB]; exists ((map_pair f)@^-1` C); first by exists C. by case=> x y [a ? ?]; apply/sBA/CsubB; exists (f a). Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent_split | |
weak_ent_nbhs : nbhs = nbhs_ weak_ent. Proof. rewrite predeq2E => x V; split. case=> [? [[B ? <-] ? BsubV]]; have: nbhs (f x) B by apply: open_nbhs_nbhs. move=> /nbhsP [W ? WsubB]; exists ((map_pair f) @^-1` W); first by exists W. by move=>??; exact/BsubV/WsubB. case=> W [V' entV' V'subW] /filterS; apply. have : nbhs (... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ent_nbhs | |
S := (weak_topology f). | Notation | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | S | |
weak_ball (x : S) (r : R) (y : S) := ball (f x) r (f y). | Definition | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ball | |
weak_pseudo_metric_ball_center (x : S) (e : R) : 0 < e -> weak_ball x e x. Proof. by move=> /posnumP[{}e]; exact: ball_center. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_pseudo_metric_ball_center | |
weak_pseudo_metric_entourageE : entourage = entourage_ weak_ball. Proof. rewrite /entourage /= /weak_ent -entourage_ballE /entourage_. have -> : (fun e => [set xy | ball (f xy.1) e (f xy.2)]) = (preimage (map_pair f) \o fun e => [set xy | ball xy.1 e xy.2])%FUN. by []. rewrite eqEsubset; split; apply/filter_fromP. - ap... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_pseudo_metric_entourageE | |
weak_ballE (e : R) (x : S) : f @^-1` (ball (f x) e) = ball x e. Proof. by []. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | weak_ballE | |
OrdU : orderTopologicalType d := order_topology (sub_type Y). | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | OrdU | |
WeakU : topologicalType := @weak_topology (sub_type Y) X val. | Let | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | WeakU | |
open_order_weak (U : set Y) : @open OrdU U -> @open WeakU U. Proof. rewrite ?openE /= /interior => + x Ux => /(_ x Ux); rewrite itv_nbhsE /=. move=> [][][[]l|[]] [[]r|[]][][]//= _ xlr /filterS; apply. - exists `]l, r[%classic; split => //=; exists `]\val l, \val r[%classic. exact: itv_open. by rewrite eqEsubset; split ... | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | open_order_weak | |
continuous_comp_weak {Y : choiceType} {X Z : topologicalType} (w : Y -> Z) (f : X -> weak_topology w) : continuous (w \o f) -> continuous f. Proof. move=> cf z U [?/= [[W oW <-]]] /= Wsfz /filterS; apply; apply: cf. exact: open_nbhs_nbhs. Qed. | Lemma | theories | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect all_algebra all_classical unstable.",
"From mathcomp Require Import interval_inference reals topology_structure.",
"From mathcomp Require Import uniform_structure order_topology.",
"From mathcomp Require Import pseudometric_st... | theories/topology_theory/weak_topology.v | continuous_comp_weak |
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