statement stringlengths 1 510 | proof stringlengths 0 12.8k | type stringclasses 24
values | symbolic_name stringlengths 1 50 | library stringclasses 14
values | filename stringclasses 114
values | imports listlengths 1 47 | deps listlengths 0 64 | docstring stringclasses 241
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
entourage_split_ex (A : set (M * M)) :
entourage A -> exists2 B, entourage B & B \; B `<=` A. | Proof. exact: entourage_split_ex_subproof. Qed. | Lemma | entourage_split_ex | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"set"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
split_ent (A : set (M * M)) | :=
get (entourage `&` [set B | B \; B `<=` A]). | Definition | split_ent | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"get",
"set"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | split_entP | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"entourage_split_ex",
"exists2P",
"getPex",
"set",
"split_ent"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
entourage_split_ent (A : set (M * M)) : entourage A ->
entourage (split_ent A). | Proof. by move=> /split_entP []. Qed. | Lemma | entourage_split_ent | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"set",
"split_ent",
"split_entP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
subset_split_ent (A : set (M * M)) : entourage A ->
split_ent A \; split_ent A `<=` A. | Proof. by move=> /split_entP []. Qed. | Lemma | subset_split_ent | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"set",
"split_ent",
"split_entP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | entourage_split | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"split_ent",
"subset_split_ent"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
nbhs_entourage (x : M) A : entourage A -> nbhs x (xsection A x). | Proof. by move=> ?; apply/nbhsP; exists A. Qed. | Lemma | nbhs_entourage | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"nbhs",
"nbhsP",
"xsection"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 apply/funext => E; apply/seteqP; split => [|] ? /xsectionP.
by rewrite filter_from_entourageE.
Qed. | Lemma | cvg_entourageP | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"Filter",
"entourage",
"filter_from",
"filter_fromP",
"filter_from_entourageE",
"funext",
"near",
"seteqP",
"split",
"xsection",
"xsectionP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | cvg_entourage | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"Filter",
"cvg_entourageP",
"entourage",
"near"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | cvg_app_entourageP | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"Filter",
"cvg_entourageP",
"entourage",
"near"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
entourage_invI (E : set (M * M)) : entourage E -> entourage (E `&` E^-1). | Proof. by move=> ?; apply: filterI; last exact: entourage_inv. Qed. | Lemma | entourage_invI | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"entourage_inv",
"set"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | split_ent_subset | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entE",
"entourage",
"entourage_refl",
"entourage_split_ent",
"set",
"split_ent",
"subset_split_ent"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
entourage_pfilter {M : puniformType} :
ProperFilter (@entourage M). | Proof.
apply Build_ProperFilter_ex; last exact: entourage_filter.
by move=> A entA; exists (point, point); apply: entourage_refl.
Qed. | Instance | entourage_pfilter | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"Build_ProperFilter_ex",
"ProperFilter",
"entourage",
"entourage_filter",
"entourage_refl"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 [_ /xsectionP yz]].
by apply/xsectionP; move: xy yz; exact: entourage_split.
Qed. | Lemma | ent_closure | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"closure",
"entE",
"entourage",
"entourage_split",
"nbhs_entourageE",
"split_ent",
"xsection",
"xsectionI",
"xsectionP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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.
by have [->|] := eqVneq y x; [by apply: nbhs_singleton|near: y].
Unshelve. all: by end_near. Qed. | Lemma | continuous_withinNx | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"continuous",
"cvg_within",
"end_near",
"nbhs_nearE",
"nbhs_singleton",
"near",
"near_map",
"near_nbhs",
"near_simpl",
"near_withinE",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 => /(_ (f y))/= AV [AVfy yx];
have /contra_neq /(_ yx) := fI y; tauto.
Qed. | Lemma | continuous_injective_withinNx | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"cf",
"continuous",
"dnbhs",
"nbhs",
"predeqP",
"seteqP",
"split",
"withinE"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | countable_uniformity | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"countable",
"entourage",
"set_system"
] | This property is primarily useful for metrizability on uniform spaces | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
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; apply: eM; exact: funS.
by move=> ? /Msub [Q + ?] => /(@surj _ _ _ _ f)[n _ fQ]; exists n; rewrite fQ.
exists (range f); split; first exact: card_image_le.
by move=> E [n ... | Lemma | countable_uniformityP | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"card_image_le",
"countable_uniformity",
"entourage",
"entourageT",
"pfcard_geP",
"range",
"set",
"split",
"surj",
"unsquash"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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 | open_nbhs_entourage | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"nbhs_entourage",
"nbhs_interior",
"nbhs_singleton",
"open_interior",
"open_nbhs",
"set",
"split",
"xsection"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
unif_continuous (U V : uniformType) (f : U -> V) | :=
(fun xy => (f xy.1, f xy.2)) @ entourage --> entourage. | Definition | unif_continuous | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
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). | Definition | entourage_set | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"set"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
cauchy {T : uniformType} (F : set_system T) | := (F, F) --> entourage. | Definition | cauchy | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"entourage",
"set_system"
] | Complete uniform spaces | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
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%relation => //; exact: entourage_inv.
move=> ab [/= /xsectionP Balima /xsectionP Blimb]; a... | Lemma | cvg_cauchy | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"Filter",
"ab",
"cauchy",
"cvg",
"entourage_inv",
"entourage_split_ex",
"lim",
"nbhs_entourageE",
"set_system",
"split",
"xsection",
"xsectionP"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
complete_ax | := cauchy_cvg (only parsing). | Notation | complete_ax | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad | |
cauchy_cvgP (F : set_system T) (FF : ProperFilter F) : cauchy F <-> cvg F. | Proof. by split=> [/cauchy_cvg|/cvg_cauchy]. Qed. | Lemma | cauchy_cvgP | theories.topology_theory | theories/topology_theory/uniform_structure.v | [
"HB",
"structures",
"mathcomp",
"all_ssreflect_compat",
"all_algebra",
"all_classical",
"topology_structure"
] | [
"ProperFilter",
"cauchy",
"cvg",
"cvg_cauchy",
"set_system",
"split"
] | https://github.com/math-comp/analysis | 723425a8e25ee4d32ff8409d0294d25d4e43f9ad |
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