statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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pi_opens_iso_sections_family : pi_opens F U ≅ Π i : ι, F.obj (op (U i)) | limits.is_limit.cone_point_unique_up_to_iso
(limit.is_limit (discrete.functor (λ i : ι, F.obj (op (U i)))))
((types.product_limit_cone.{v v} (λ i : ι, F.obj (op (U i)))).is_limit) | def | Top.presheaf.pi_opens_iso_sections_family | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | For presheaves of types, terms of `pi_opens F U` are just families of sections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compatible_iff_left_res_eq_right_res (sf : pi_opens F U) :
is_compatible F U ((pi_opens_iso_sections_family F U).hom sf)
↔ left_res F U sf = right_res F U sf | begin
split ; intros h,
{ ext ⟨i, j⟩,
rw [left_res, types.limit.lift_π_apply', fan.mk_π_app,
right_res, types.limit.lift_π_apply', fan.mk_π_app],
exact h i j, },
{ intros i j,
convert congr_arg (limits.pi.π (λ p : ι × ι, F.obj (op (U p.1 ⊓ U p.2))) (i,j)) h,
{ rw [left_res, types.pi_lift_π... | lemma | Top.presheaf.compatible_iff_left_res_eq_right_res | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | Under the isomorphism `pi_opens_iso_sections_family`, compatibility of sections is the same
as being equalized by the arrows `left_res` and `right_res` of the equalizer diagram. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_gluing_iff_eq_res (sf : pi_opens F U) (s : F.obj (op (supr U))):
is_gluing F U ((pi_opens_iso_sections_family F U).hom sf) s ↔ res F U s = sf | begin
split ; intros h,
{ ext ⟨i⟩,
rw [res, types.limit.lift_π_apply', fan.mk_π_app],
exact h i, },
{ intro i,
convert congr_arg (limits.pi.π (λ i : ι, F.obj (op (U i))) i) h,
rw [res, types.pi_lift_π_apply],
refl },
end | lemma | Top.presheaf.is_gluing_iff_eq_res | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr"
] | Under the isomorphism `pi_opens_iso_sections_family`, being a gluing of a family of
sections `sf` is the same as lying in the preimage of `res` (the leftmost arrow of the
equalizer diagram). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_of_is_sheaf_unique_gluing_types (Fsh : F.is_sheaf_unique_gluing) :
F.is_sheaf | begin
rw is_sheaf_iff_is_sheaf_equalizer_products,
intros ι U,
refine ⟨fork.is_limit.mk' _ _⟩,
intro s,
have h_compatible : ∀ x : s.X,
F.is_compatible U ((F.pi_opens_iso_sections_family U).hom (s.ι x)),
{ intro x,
rw compatible_iff_left_res_eq_right_res,
convert congr_fun s.condition x, },
cho... | lemma | Top.presheaf.is_sheaf_of_is_sheaf_unique_gluing_types | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | The "equalizer" sheaf condition can be obtained from the sheaf condition
in terms of unique gluings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_unique_gluing_of_is_sheaf_types (Fsh : F.is_sheaf) :
F.is_sheaf_unique_gluing | begin
rw is_sheaf_iff_is_sheaf_equalizer_products at Fsh,
intros ι U sf hsf,
let sf' := (pi_opens_iso_sections_family F U).inv sf,
have hsf' : left_res F U sf' = right_res F U sf',
{ rwa [← compatible_iff_left_res_eq_right_res F U sf', inv_hom_id_apply] },
choose s s_spec s_uniq using types.unique_of_type_e... | lemma | Top.presheaf.is_sheaf_unique_gluing_of_is_sheaf_types | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | The sheaf condition in terms of unique gluings can be obtained from the usual
"equalizer" sheaf condition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_unique_gluing_types :
F.is_sheaf ↔ F.is_sheaf_unique_gluing | iff.intro (is_sheaf_unique_gluing_of_is_sheaf_types F)
(is_sheaf_of_is_sheaf_unique_gluing_types F) | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_unique_gluing_types | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | For type-valued presheaves, the sheaf condition in terms of unique gluings is equivalent to the
usual sheaf condition in terms of equalizer diagrams. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_sheaf_iff_is_sheaf_unique_gluing :
F.is_sheaf ↔ F.is_sheaf_unique_gluing | iff.trans (is_sheaf_iff_is_sheaf_comp (forget C) F)
(is_sheaf_iff_is_sheaf_unique_gluing_types (F ⋙ forget C)) | lemma | Top.presheaf.is_sheaf_iff_is_sheaf_unique_gluing | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [] | For presheaves valued in a concrete category, whose forgetful functor reflects isomorphisms and
preserves limits, the sheaf condition in terms of unique gluings is equivalent to the usual one
in terms of equalizer diagrams. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_unique_gluing (sf : Π i : ι, F.1.obj (op (U i)))
(h : is_compatible F.1 U sf ) :
∃! s : F.1.obj (op (supr U)), is_gluing F.1 U sf s | (is_sheaf_iff_is_sheaf_unique_gluing F.1).mp F.cond U sf h | lemma | Top.sheaf.exists_unique_gluing | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr"
] | A more convenient way of obtaining a unique gluing of sections for a sheaf. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_unique_gluing' (V : opens X) (iUV : Π i : ι, U i ⟶ V) (hcover : V ≤ supr U)
(sf : Π i : ι, F.1.obj (op (U i))) (h : is_compatible F.1 U sf) :
∃! s : F.1.obj (op V), ∀ i : ι, F.1.map (iUV i).op s = sf i | begin
have V_eq_supr_U : V = supr U := le_antisymm hcover (supr_le (λ i, (iUV i).le)),
obtain ⟨gl, gl_spec, gl_uniq⟩ := F.exists_unique_gluing U sf h,
refine ⟨F.1.map (eq_to_hom V_eq_supr_U).op gl, _, _⟩,
{ intro i,
rw [← comp_apply, ← F.1.map_comp],
exact gl_spec i },
{ intros gl' gl'_spec,
conve... | lemma | Top.sheaf.exists_unique_gluing' | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr",
"supr_le"
] | In this version of the lemma, the inclusion homs `iUV` can be specified directly by the user,
which can be more convenient in practice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_locally_eq (s t : F.1.obj (op (supr U)))
(h : ∀ i, F.1.map (opens.le_supr U i).op s = F.1.map (opens.le_supr U i).op t) :
s = t | begin
let sf : Π i : ι, F.1.obj (op (U i)) := λ i, F.1.map (opens.le_supr U i).op s,
have sf_compatible : is_compatible _ U sf,
{ intros i j, simp_rw [← comp_apply, ← F.1.map_comp], refl },
obtain ⟨gl, -, gl_uniq⟩ := F.exists_unique_gluing U sf sf_compatible,
transitivity gl,
{ apply gl_uniq, intro i, refl ... | lemma | Top.sheaf.eq_of_locally_eq | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_locally_eq' (V : opens X) (iUV : Π i : ι, U i ⟶ V) (hcover : V ≤ supr U)
(s t : F.1.obj (op V))
(h : ∀ i, F.1.map (iUV i).op s = F.1.map (iUV i).op t) : s = t | begin
have V_eq_supr_U : V = supr U := le_antisymm hcover (supr_le (λ i, (iUV i).le)),
suffices : F.1.map (eq_to_hom V_eq_supr_U.symm).op s =
F.1.map (eq_to_hom V_eq_supr_U.symm).op t,
{ convert congr_arg (F.1.map (eq_to_hom V_eq_supr_U).op) this ;
rw [← comp_apply, ← F.1.map_comp, eq_to_hom_op, ... | lemma | Top.sheaf.eq_of_locally_eq' | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"supr",
"supr_le"
] | In this version of the lemma, the inclusion homs `iUV` can be specified directly by the user,
which can be more convenient in practice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_locally_eq₂ {U₁ U₂ V : opens X}
(i₁ : U₁ ⟶ V) (i₂ : U₂ ⟶ V) (hcover : V ≤ U₁ ⊔ U₂)
(s t : F.1.obj (op V))
(h₁ : F.1.map i₁.op s = F.1.map i₁.op t)
(h₂ : F.1.map i₂.op s = F.1.map i₂.op t) : s = t | begin
classical,
fapply F.eq_of_locally_eq' (λ t : ulift bool, if t.1 then U₁ else U₂),
{ exact λ i, if h : i.1 then (eq_to_hom (if_pos h)) ≫ i₁ else (eq_to_hom (if_neg h)) ≫ i₂ },
{ refine le_trans hcover _, rw sup_le_iff, split,
{ convert le_supr (λ t : ulift bool, if t.1 then U₁ else U₂) (ulift.up true) ... | lemma | Top.sheaf.eq_of_locally_eq₂ | topology.sheaves.sheaf_condition | src/topology/sheaves/sheaf_condition/unique_gluing.lean | [
"topology.sheaves.forget",
"category_theory.limits.shapes.types",
"topology.sheaves.sheaf",
"category_theory.types"
] | [
"le_supr",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_spectral_map (f : α → β) extends continuous f : Prop | (is_compact_preimage_of_is_open ⦃s : set β⦄ : is_open s → is_compact s → is_compact (f ⁻¹' s)) | structure | is_spectral_map | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"continuous",
"is_compact",
"is_open"
] | A function between topological spaces is spectral if it is continuous and the preimage of every
compact open set is compact open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.preimage_of_is_open (hf : is_spectral_map f) (h₀ : is_compact s) (h₁ : is_open s) :
is_compact (f ⁻¹' s) | hf.is_compact_preimage_of_is_open h₁ h₀ | lemma | is_compact.preimage_of_is_open | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"is_compact",
"is_open",
"is_spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_spectral_map.continuous {f : α → β} (hf : is_spectral_map f) : continuous f | hf.to_continuous | lemma | is_spectral_map.continuous | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"continuous",
"is_spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_spectral_map_id : is_spectral_map (@id α) | ⟨continuous_id, λ s _, id⟩ | lemma | is_spectral_map_id | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"is_spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_spectral_map.comp {f : β → γ} {g : α → β} (hf : is_spectral_map f)
(hg : is_spectral_map g) :
is_spectral_map (f ∘ g) | ⟨hf.continuous.comp hg.continuous,
λ s hs₀ hs₁, (hs₁.preimage_of_is_open hf hs₀).preimage_of_is_open hg (hs₀.preimage hf.continuous)⟩ | lemma | is_spectral_map.comp | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"is_spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectral_map (α β : Type*) [topological_space α] [topological_space β] | (to_fun : α → β)
(spectral' : is_spectral_map to_fun) | structure | spectral_map | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"is_spectral_map",
"topological_space"
] | The type of spectral maps from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spectral_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[topological_space β]
extends fun_like F α (λ _, β) | (map_spectral (f : F) : is_spectral_map f) | class | spectral_map_class | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"fun_like",
"is_spectral_map",
"topological_space"
] | `spectral_map_class F α β` states that `F` is a type of spectral maps.
You should extend this class when you extend `spectral_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spectral_map_class.to_continuous_map_class [topological_space α] [topological_space β]
[spectral_map_class F α β] :
continuous_map_class F α β | { map_continuous := λ f, (map_spectral f).continuous,
..‹spectral_map_class F α β› } | instance | spectral_map_class.to_continuous_map_class | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"continuous",
"continuous_map_class",
"spectral_map_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (f : spectral_map α β) : continuous_map α β | ⟨_, f.spectral'.continuous⟩ | def | spectral_map.to_continuous_map | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map",
"spectral_map"
] | Reinterpret a `spectral_map` as a `continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : spectral_map α β} : f.to_fun = (f : α → β) | rfl | lemma | spectral_map.to_fun_eq_coe | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : spectral_map α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | spectral_map.ext | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"fun_like.ext",
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : spectral_map α β) (f' : α → β) (h : f' = f) : spectral_map α β | ⟨f', h.symm.subst f.spectral'⟩ | def | spectral_map.copy | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | Copy of a `spectral_map` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : spectral_map α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | spectral_map.coe_copy | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : spectral_map α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | spectral_map.copy_eq | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"fun_like.ext'",
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : spectral_map α α | ⟨id, is_spectral_map_id⟩ | def | spectral_map.id | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | `id` as a `spectral_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(spectral_map.id α) = id | rfl | lemma | spectral_map.coe_id | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : spectral_map.id α a = a | rfl | lemma | spectral_map.id_apply | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : spectral_map β γ) (g : spectral_map α β) : spectral_map α γ | ⟨f.to_continuous_map.comp g.to_continuous_map, f.spectral'.comp g.spectral'⟩ | def | spectral_map.comp | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | Composition of `spectral_map`s as a `spectral_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : spectral_map β γ) (g : spectral_map α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | spectral_map.coe_comp | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : spectral_map β γ) (g : spectral_map α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | spectral_map.comp_apply | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_continuous_map (f : spectral_map β γ) (g : spectral_map α β) :
(f.comp g : continuous_map α γ) = (f : continuous_map β γ).comp g | rfl | lemma | spectral_map.coe_comp_continuous_map | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map",
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : spectral_map γ δ) (g : spectral_map β γ) (h : spectral_map α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | spectral_map.comp_assoc | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : spectral_map α β) : f.comp (spectral_map.id α) = f | ext $ λ a, rfl | lemma | spectral_map.comp_id | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map",
"spectral_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : spectral_map α β) : (spectral_map.id β).comp f = f | ext $ λ a, rfl | lemma | spectral_map.id_comp | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map",
"spectral_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : spectral_map β γ} {f : spectral_map α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | spectral_map.cancel_right | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : spectral_map β γ} {f₁ f₂ : spectral_map α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | spectral_map.cancel_left | topology.spectral | src/topology/spectral/hom.lean | [
"topology.continuous_function.basic"
] | [
"spectral_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space : uniform_space R | uniform_space.of_fun (λ x y, abv (y - x)) (by simp) (λ x y, abv.map_sub y x)
(λ x y z, (abv.sub_le _ _ _).trans_eq (add_comm _ _)) $
λ ε ε0, ⟨ε / 2, half_pos ε0, λ _ h₁ _ h₂, (add_lt_add h₁ h₂).trans_eq (add_halves ε)⟩ | def | absolute_value.uniform_space | topology.uniform_space | src/topology/uniform_space/absolute_value.lean | [
"algebra.order.absolute_value",
"topology.uniform_space.basic"
] | [
"add_halves",
"half_pos",
"uniform_space",
"uniform_space.of_fun"
] | The uniform space structure coming from an absolute value. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity :
𝓤[abv.uniform_space].has_basis (λ ε : 𝕜, 0 < ε) (λ ε, {p : R × R | abv (p.2 - p.1) < ε}) | uniform_space.has_basis_of_fun (exists_gt _) _ _ _ _ _ | theorem | absolute_value.has_basis_uniformity | topology.uniform_space | src/topology/uniform_space/absolute_value.lean | [
"algebra.order.absolute_value",
"topology.uniform_space.basic"
] | [
"uniform_space.has_basis_of_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abstract_completion (α : Type u) [uniform_space α] | (space : Type u)
(coe : α → space)
(uniform_struct : uniform_space space)
(complete : complete_space space)
(separation : separated_space space)
(uniform_inducing : uniform_inducing coe)
(dense : dense_range coe) | structure | abstract_completion | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"complete_space",
"dense",
"dense_range",
"separated_space",
"uniform_inducing",
"uniform_space"
] | A completion of `α` is the data of a complete separated uniform space (from the same universe)
and a map from `α` with dense range and inducing the original uniform structure on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_complete [separated_space α] [complete_space α] : abstract_completion α | mk α id infer_instance infer_instance infer_instance uniform_inducing_id dense_range_id | def | abstract_completion.of_complete | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"abstract_completion",
"complete_space",
"dense_range_id",
"separated_space",
"uniform_inducing_id"
] | If `α` is complete, then it is an abstract completion of itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_range : closure (range ι) = univ | pkg.dense.closure_range | lemma | abstract_completion.closure_range | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_inducing : dense_inducing ι | ⟨pkg.uniform_inducing.inducing, pkg.dense⟩ | lemma | abstract_completion.dense_inducing | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_coe : uniform_continuous ι | uniform_inducing.uniform_continuous pkg.uniform_inducing | lemma | abstract_completion.uniform_continuous_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_inducing.uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous ι | pkg.uniform_continuous_coe.continuous | lemma | abstract_completion.continuous_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {p : hatα → Prop}
(a : hatα) (hp : is_closed {a | p a}) (ih : ∀ a, p (ι a)) : p a | is_closed_property pkg.dense hp ih a | lemma | abstract_completion.induction_on | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"ih",
"is_closed",
"is_closed_property"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
funext [topological_space β] [t2_space β] {f g : hatα → β}
(hf : continuous f) (hg : continuous g)
(h : ∀ a, f (ι a) = g (ι a)) : f = g | funext $ assume a, pkg.induction_on a (is_closed_eq hf hg) h | lemma | abstract_completion.funext | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous",
"is_closed_eq",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend (f : α → β) : hatα → β | if uniform_continuous f then
pkg.dense_inducing.extend f
else
λ x, f (pkg.dense.some x) | def | abstract_completion.extend | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"extend",
"uniform_continuous"
] | Extension of maps to completions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_def (hf : uniform_continuous f) : pkg.extend f = pkg.dense_inducing.extend f | if_pos hf | lemma | abstract_completion.extend_def | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"extend_def",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_coe [t2_space β] (hf : uniform_continuous f) (a : α) :
(pkg.extend f) (ι a) = f a | begin
rw pkg.extend_def hf,
exact pkg.dense_inducing.extend_eq hf.continuous a
end | lemma | abstract_completion.extend_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"t2_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_extend : uniform_continuous (pkg.extend f) | begin
by_cases hf : uniform_continuous f,
{ rw pkg.extend_def hf,
exact uniform_continuous_uniformly_extend (pkg.uniform_inducing)
(pkg.dense) hf },
{ change uniform_continuous (ite _ _ _),
rw if_neg hf,
exact uniform_continuous_of_const (assume a b, by congr) }
end | lemma | abstract_completion.uniform_continuous_extend | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_of_const",
"uniform_continuous_uniformly_extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_extend : continuous (pkg.extend f) | pkg.uniform_continuous_extend.continuous | lemma | abstract_completion.continuous_extend | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_unique (hf : uniform_continuous f) {g : hatα → β} (hg : uniform_continuous g)
(h : ∀ a : α, f a = g (ι a)) : pkg.extend f = g | begin
apply pkg.funext pkg.continuous_extend hg.continuous,
simpa only [pkg.extend_coe hf] using h
end | lemma | abstract_completion.extend_unique | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_comp_coe {f : hatα → β} (hf : uniform_continuous f) :
pkg.extend (f ∘ ι) = f | funext $ λ x, pkg.induction_on x (is_closed_eq pkg.continuous_extend hf.continuous)
(λ y, pkg.extend_coe (hf.comp $ pkg.uniform_continuous_coe) y) | lemma | abstract_completion.extend_comp_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"is_closed_eq",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) : hatα → hatβ | pkg.extend (ι' ∘ f) | def | abstract_completion.map | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [] | Lifting maps to completions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_map : uniform_continuous (map f) | pkg.uniform_continuous_extend | lemma | abstract_completion.uniform_continuous_map | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map : continuous (map f) | pkg.continuous_extend | lemma | abstract_completion.continuous_map | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous",
"continuous_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe (hf : uniform_continuous f) (a : α) : map f (ι a) = ι' (f a) | pkg.extend_coe (pkg'.uniform_continuous_coe.comp hf) a | lemma | abstract_completion.map_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unique {f : α → β} {g : hatα → hatβ}
(hg : uniform_continuous g) (h : ∀ a, ι' (f a) = g (ι a)) : map f = g | pkg.funext (pkg.continuous_map _ _) hg.continuous $
begin
intro a,
change pkg.extend (ι' ∘ f) _ = _,
simp only [(∘), h],
rw [pkg.extend_coe (hg.comp pkg.uniform_continuous_coe)]
end | lemma | abstract_completion.map_unique | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : pkg.map pkg id = id | pkg.map_unique pkg uniform_continuous_id (assume a, rfl) | lemma | abstract_completion.map_id | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"map_id",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_map [complete_space γ] [separated_space γ] {f : β → γ} {g : α → β}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
pkg'.extend f ∘ map g = pkg.extend (f ∘ g) | pkg.funext (pkg'.continuous_extend.comp (pkg.continuous_map pkg' _)) pkg.continuous_extend $ λ a,
by rw [pkg.extend_coe (hf.comp hg), comp_app, pkg.map_coe pkg' hg, pkg'.extend_coe hf] | lemma | abstract_completion.extend_map | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"complete_space",
"separated_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {g : β → γ} {f : α → β} (hg : uniform_continuous g) (hf : uniform_continuous f) :
(pkg'.map pkg'' g) ∘ (pkg.map pkg' f) = pkg.map pkg'' (g ∘ f) | pkg.extend_map pkg' (pkg''.uniform_continuous_coe.comp hg) hf | lemma | abstract_completion.map_comp | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"map_comp",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compare : pkg.space → pkg'.space | pkg.extend pkg'.coe | def | abstract_completion.compare | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [] | The comparison map between two completions of the same uniform space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_compare : uniform_continuous (pkg.compare pkg') | pkg.uniform_continuous_extend | lemma | abstract_completion.uniform_continuous_compare | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compare_coe (a : α) : pkg.compare pkg' (pkg.coe a) = pkg'.coe a | pkg.extend_coe pkg'.uniform_continuous_coe a | lemma | abstract_completion.compare_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse_compare : (pkg.compare pkg') ∘ (pkg'.compare pkg) = id | begin
have uc := pkg.uniform_continuous_compare pkg',
have uc' := pkg'.uniform_continuous_compare pkg,
apply pkg'.funext (uc.comp uc').continuous continuous_id,
intro a,
rw [comp_app, pkg'.compare_coe pkg, pkg.compare_coe pkg'],
refl
end | lemma | abstract_completion.inverse_compare | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compare_equiv : pkg.space ≃ᵤ pkg'.space | { to_fun := pkg.compare pkg',
inv_fun := pkg'.compare pkg,
left_inv := congr_fun (pkg'.inverse_compare pkg),
right_inv := congr_fun (pkg.inverse_compare pkg'),
uniform_continuous_to_fun := uniform_continuous_compare _ _,
uniform_continuous_inv_fun := uniform_continuous_compare _ _, } | def | abstract_completion.compare_equiv | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"inv_fun"
] | The uniform bijection between two completions of the same uniform space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_compare_equiv : uniform_continuous (pkg.compare_equiv pkg') | pkg.uniform_continuous_compare pkg' | lemma | abstract_completion.uniform_continuous_compare_equiv | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_compare_equiv_symm : uniform_continuous (pkg.compare_equiv pkg').symm | pkg'.uniform_continuous_compare pkg | lemma | abstract_completion.uniform_continuous_compare_equiv_symm | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : abstract_completion (α × β) | { space := hatα × hatβ,
coe := λ p, ⟨ι p.1, ι' p.2⟩,
uniform_struct := prod.uniform_space,
complete := by apply_instance,
separation := by apply_instance,
uniform_inducing := uniform_inducing.prod pkg.uniform_inducing pkg'.uniform_inducing,
dense := pkg.dense.prod_map pkg'.dense } | def | abstract_completion.prod | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"abstract_completion",
"dense",
"uniform_inducing",
"uniform_inducing.prod"
] | Products of completions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend₂ (f : α → β → γ) : hatα → hatβ → γ | curry $ (pkg.prod pkg').extend (uncurry f) | def | abstract_completion.extend₂ | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"extend"
] | Extend two variable map to completions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension₂_coe_coe (hf : uniform_continuous $ uncurry f) (a : α) (b : β) :
pkg.extend₂ pkg' f (ι a) (ι' b) = f a b | show (pkg.prod pkg').extend (uncurry f) ((pkg.prod pkg').coe (a, b)) = uncurry f (a, b),
from (pkg.prod pkg').extend_coe hf _ | lemma | abstract_completion.extension₂_coe_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"extend",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_extension₂ : uniform_continuous₂ (pkg.extend₂ pkg' f) | begin
rw [uniform_continuous₂_def, abstract_completion.extend₂, uncurry_curry],
apply uniform_continuous_extend
end | lemma | abstract_completion.uniform_continuous_extension₂ | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"abstract_completion.extend₂",
"uniform_continuous₂",
"uniform_continuous₂_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂ (f : α → β → γ) : hatα → hatβ → hatγ | pkg.extend₂ pkg' (pkg''.coe ∘₂ f) | def | abstract_completion.map₂ | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [] | Lift two variable maps to completions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_map₂ (f : α → β → γ) : uniform_continuous₂ (pkg.map₂ pkg' pkg'' f) | pkg.uniform_continuous_extension₂ pkg' _ | lemma | abstract_completion.uniform_continuous_map₂ | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map₂ {δ} [topological_space δ] {f : α → β → γ}
{a : δ → hatα} {b : δ → hatβ} (ha : continuous a) (hb : continuous b) :
continuous (λd:δ, pkg.map₂ pkg' pkg'' f (a d) (b d)) | ((pkg.uniform_continuous_map₂ pkg' pkg'' f).continuous.comp (continuous.prod_mk ha hb) : _) | lemma | abstract_completion.continuous_map₂ | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"continuous",
"continuous.comp",
"continuous.prod_mk",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : uniform_continuous₂ f) :
pkg.map₂ pkg' pkg'' f (ι a) (ι' b) = ι'' (f a b) | pkg.extension₂_coe_coe pkg' (pkg''.uniform_continuous_coe.comp hf) a b | lemma | abstract_completion.map₂_coe_coe | topology.uniform_space | src/topology/uniform_space/abstract_completion.lean | [
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.equiv"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_rel {α : Type*} | {p : α × α | p.1 = p.2} | def | id_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | The identity relation, or the graph of the identity function | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_id_rel {a b : α} : (a, b) ∈ @id_rel α ↔ a = b | iff.rfl | theorem | mem_id_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_rel_subset {s : set (α × α)} : id_rel ⊆ s ↔ ∀ a, (a, a) ∈ s | by simp [subset_def]; exact forall_congr (λ a, by simp) | theorem | id_rel_subset | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_rel {α : Type u} (r₁ r₂ : set (α×α)) | {p : α × α | ∃z:α, (p.1, z) ∈ r₁ ∧ (z, p.2) ∈ r₂} | def | comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | The composition of relations | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_comp_rel {r₁ r₂ : set (α×α)}
{x y : α} : (x, y) ∈ r₁ ○ r₂ ↔ ∃ z, (x, z) ∈ r₁ ∧ (z, y) ∈ r₂ | iff.rfl | theorem | mem_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_id_rel : prod.swap '' id_rel = @id_rel α | set.ext $ assume ⟨a, b⟩, by simp [image_swap_eq_preimage_swap]; exact eq_comm | theorem | swap_id_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel",
"prod.swap",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone.comp_rel [preorder β] {f g : β → set (α×α)}
(hf : monotone f) (hg : monotone g) : monotone (λx, (f x) ○ (g x)) | assume a b h p ⟨z, h₁, h₂⟩, ⟨z, hf h h₁, hg h h₂⟩ | theorem | monotone.comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_rel_mono {f g h k: set (α×α)} (h₁ : f ⊆ h) (h₂ : g ⊆ k) : f ○ g ⊆ h ○ k | λ ⟨x, y⟩ ⟨z, h, h'⟩, ⟨z, h₁ h, h₂ h'⟩ | lemma | comp_rel_mono | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk_mem_comp_rel {a b c : α} {s t : set (α×α)} (h₁ : (a, c) ∈ s) (h₂ : (c, b) ∈ t) :
(a, b) ∈ s ○ t | ⟨c, h₁, h₂⟩ | lemma | prod_mk_mem_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp_rel {r : set (α×α)} : id_rel ○ r = r | set.ext $ assume ⟨a, b⟩, by simp | lemma | id_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_rel_assoc {r s t : set (α×α)} :
(r ○ s) ○ t = r ○ (s ○ t) | by ext p; cases p; simp only [mem_comp_rel]; tauto | lemma | comp_rel_assoc | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"mem_comp_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_subset_comp_rel {s t : set (α × α)} (h : id_rel ⊆ t) : s ⊆ s ○ t | λ ⟨x, y⟩ xy_in, ⟨y, xy_in, h $ by exact rfl⟩ | lemma | left_subset_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_subset_comp_rel {s t : set (α × α)} (h : id_rel ⊆ s) : t ⊆ s ○ t | λ ⟨x, y⟩ xy_in, ⟨x, h $ by exact rfl, xy_in⟩ | lemma | right_subset_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_comp_self {s : set (α × α)} (h : id_rel ⊆ s) : s ⊆ s ○ s | left_subset_comp_rel h | lemma | subset_comp_self | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel",
"left_subset_comp_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_iterate_comp_rel {s t : set (α × α)} (h : id_rel ⊆ s) (n : ℕ) :
t ⊆ (((○) s) ^[n] t) | begin
induction n with n ihn generalizing t,
exacts [subset.rfl, (right_subset_comp_rel h).trans ihn]
end | lemma | subset_iterate_comp_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"id_rel",
"right_subset_comp_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetric_rel (V : set (α × α)) : Prop | prod.swap ⁻¹' V = V | def | symmetric_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap"
] | The relation is invariant under swapping factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmetrize_rel (V : set (α × α)) : set (α × α) | V ∩ prod.swap ⁻¹' V | def | symmetrize_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap"
] | The maximal symmetric relation contained in a given relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmetric_symmetrize_rel (V : set (α × α)) : symmetric_rel (symmetrize_rel V) | by simp [symmetric_rel, symmetrize_rel, preimage_inter, inter_comm, ← preimage_comp] | lemma | symmetric_symmetrize_rel | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetric_rel",
"symmetrize_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetrize_rel_subset_self (V : set (α × α)) : symmetrize_rel V ⊆ V | sep_subset _ _ | lemma | symmetrize_rel_subset_self | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetrize_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetrize_mono {V W: set (α × α)} (h : V ⊆ W) : symmetrize_rel V ⊆ symmetrize_rel W | inter_subset_inter h $ preimage_mono h | lemma | symmetrize_mono | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetrize_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetric_rel.mk_mem_comm {V : set (α × α)} (hV : symmetric_rel V) {x y : α} :
(x, y) ∈ V ↔ (y, x) ∈ V | set.ext_iff.1 hV (y, x) | lemma | symmetric_rel.mk_mem_comm | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symmetric_rel.eq {U : set (α × α)} (hU : symmetric_rel U) : prod.swap ⁻¹' U = U | hU | lemma | symmetric_rel.eq | topology.uniform_space | src/topology/uniform_space/basic.lean | [
"order.filter.small_sets",
"topology.subset_properties",
"topology.nhds_set"
] | [
"prod.swap",
"symmetric_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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