fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
eq_of_locally_eq₂ {U₁ U₂ V : Opens X} (i₁ : U₁ ⟶ V) (i₂ : U₂ ⟶ V) (hcover : V ≤ U₁ ⊔ U₂)
(s t : ToType (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 := by
classical
fapply F.eq_of_locally_eq' fun t : Bool => if t then U₁ else U₂
· exact fun i => if h : i then eqToHom (if_pos h) ≫ i₁ else eqToHom (if_neg h) ≫ i₂
· refine le_trans hcover ?_
rw [sup_le_iff]
constructor
· exact le_iSup (fun t : Bool => if t then U₁ else U₂) true
· exact le_iSup (fun t : Bool => if t then U₁ else U₂) false
· rintro ⟨_ | _⟩
any_goals exact h₁
any_goals exact h₂
|
theorem
|
Topology
|
[
"Mathlib.Topology.Sheaves.Forget",
"Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections",
"Mathlib.CategoryTheory.Limits.Types.Shapes"
] |
Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean
|
eq_of_locally_eq₂
| null |
IsTransitiveRel (V : Set (X × X)) : Prop :=
∀ ⦃x y z⦄, (x, y) ∈ V → (y, z) ∈ V → (x, z) ∈ V
|
def
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel
|
The relation is transitive.
|
IsTransitiveRel.comp_subset_self {s : Set (X × X)}
(h : IsTransitiveRel s) :
s ○ s ⊆ s :=
fun ⟨_, _⟩ ⟨_, hxz, hzy⟩ ↦ h hxz hzy
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.comp_subset_self
| null |
isTransitiveRel_iff_comp_subset_self {s : Set (X × X)} :
IsTransitiveRel s ↔ s ○ s ⊆ s :=
⟨IsTransitiveRel.comp_subset_self, fun h _ _ _ hx hy ↦ h ⟨_, hx, hy⟩⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isTransitiveRel_iff_comp_subset_self
| null |
isTransitiveRel_empty : IsTransitiveRel (X := X) ∅ := by
simp [IsTransitiveRel]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isTransitiveRel_empty
| null |
isTransitiveRel_idRel : IsTransitiveRel (idRel : Set (X × X)) := by
simp [IsTransitiveRel, idRel]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isTransitiveRel_idRel
| null |
isTransitiveRel_univ : IsTransitiveRel (X := X) Set.univ := by
simp [IsTransitiveRel]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isTransitiveRel_univ
| null |
isTransitiveRel_singleton (x y : X) : IsTransitiveRel {(x, y)} := by
simp +contextual [IsTransitiveRel]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isTransitiveRel_singleton
| null |
IsTransitiveRel.inter {s t : Set (X × X)} (hs : IsTransitiveRel s) (ht : IsTransitiveRel t) :
IsTransitiveRel (s ∩ t) :=
fun _ _ _ h h' ↦ ⟨hs h.left h'.left, ht h.right h'.right⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.inter
| null |
IsTransitiveRel.iInter {ι : Type*} {U : (i : ι) → Set (X × X)}
(hU : ∀ i, IsTransitiveRel (U i)) :
IsTransitiveRel (⋂ i, U i) := by
intro _ _ _ h h'
simp only [mem_iInter] at h h' ⊢
intro i
exact hU i (h i) (h' i)
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.iInter
| null |
IsTransitiveRel.sInter {s : Set (Set (X × X))} (h : ∀ i ∈ s, IsTransitiveRel i) :
IsTransitiveRel (⋂₀ s) := by
rw [sInter_eq_iInter]
exact IsTransitiveRel.iInter (by simpa)
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.sInter
| null |
IsTransitiveRel.preimage_prodMap {Y : Type*} {t : Set (Y × Y)}
(ht : IsTransitiveRel t) (f : X → Y) :
IsTransitiveRel (Prod.map f f ⁻¹' t) :=
fun _ _ _ h h' ↦ ht h h'
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.preimage_prodMap
| null |
IsTransitiveRel.symmetrizeRel {s : Set (X × X)}
(h : IsTransitiveRel s) :
IsTransitiveRel (symmetrizeRel s) :=
fun _ _ _ hxy hyz ↦ ⟨h hxy.1 hyz.1, h hyz.2 hxy.2⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.symmetrizeRel
| null |
IsTransitiveRel.comp_eq_of_idRel_subset {s : Set (X × X)}
(h : IsTransitiveRel s) (h' : idRel ⊆ s) :
s ○ s = s :=
le_antisymm h.comp_subset_self (subset_comp_self h')
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.comp_eq_of_idRel_subset
| null |
IsTransitiveRel.prod_subset_trans {s : Set (X × X)} {t u v : Set X} (hs : IsTransitiveRel s)
(htu : t ×ˢ u ⊆ s) (huv : u ×ˢ v ⊆ s) (hu : u.Nonempty) :
t ×ˢ v ⊆ s := by
rintro ⟨a, b⟩ hab
simp only [mem_prod] at hab
obtain ⟨x, hx⟩ := hu
exact hs (@htu ⟨a, x⟩ ⟨hab.left, hx⟩) (@huv ⟨x, b⟩ ⟨hx, hab.right⟩)
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.prod_subset_trans
| null |
IsTransitiveRel.mem_filter_prod_trans {s : Set (X × X)} {f g h : Filter X} [g.NeBot]
(hs : IsTransitiveRel s) (hfg : s ∈ f ×ˢ g) (hgh : s ∈ g ×ˢ h) :
s ∈ f ×ˢ h :=
Eventually.trans_prod (by simpa using hfg) (by simpa using hgh) hs
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.mem_filter_prod_trans
| null |
IsTransitiveRel.mem_filter_prod_comm {s : Set (X × X)} {f g h : Filter X} [g.NeBot]
(hs : IsTransitiveRel s) (hfg : s ∈ f ×ˢ g) (hgh : s ∈ g ×ˢ h) :
s ∈ f ×ˢ h := by
rw [mem_prod_iff] at hfg hgh ⊢
obtain ⟨t, ht, u, hu, htu⟩ := hfg
obtain ⟨v, hv, w, hw, hvw⟩ := hgh
replace htu : t ×ˢ (u ∩ v) ⊆ s := by
rw [Set.prod_inter]
refine inter_subset_left.trans htu
replace hvw : (u ∩ v) ×ˢ w ⊆ s := by
rw [Set.inter_prod]
refine inter_subset_right.trans hvw
refine ⟨_, ht, _, hw, hs.prod_subset_trans htu hvw <| g.nonempty_of_mem ?_⟩
simp [hu, hv]
open UniformSpace in
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.mem_filter_prod_comm
| null |
IsTransitiveRel.ball_subset_of_mem {V : Set (X × X)} (h : IsTransitiveRel V)
{x y : X} (hy : y ∈ ball x V) :
ball y V ⊆ ball x V :=
ball_subset_of_comp_subset hy (h.comp_subset_self)
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsTransitiveRel.ball_subset_of_mem
| null |
UniformSpace.ball_eq_of_mem_of_isSymmetricRel_of_isTransitiveRel {V : Set (X × X)}
(h_symm : IsSymmetricRel V) (h_trans : IsTransitiveRel V) {x y : X}
(hy : y ∈ ball x V) :
ball x V = ball y V := by
refine le_antisymm (h_trans.ball_subset_of_mem ?_) (h_trans.ball_subset_of_mem hy)
rwa [← mem_ball_symmetry h_symm]
variable [UniformSpace X]
variable (X) in
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
UniformSpace.ball_eq_of_mem_of_isSymmetricRel_of_isTransitiveRel
| null |
IsUltraUniformity : Prop where
hasBasis : (𝓤 X).HasBasis
(fun s : Set (X × X) => s ∈ 𝓤 X ∧ IsSymmetricRel s ∧ IsTransitiveRel s) id
|
class
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsUltraUniformity
|
A uniform space is ultrametric if the uniformity `𝓤 X` has a basis of equivalence relations.
|
IsUltraUniformity.mk_of_hasBasis {ι : Type*} {p : ι → Prop} {s : ι → Set (X × X)}
(h_basis : (𝓤 X).HasBasis p s) (h_symm : ∀ i, p i → IsSymmetricRel (s i))
(h_trans : ∀ i, p i → IsTransitiveRel (s i)) :
IsUltraUniformity X where
hasBasis := h_basis.to_hasBasis'
(fun i hi ↦ ⟨s i, ⟨h_basis.mem_of_mem hi, h_symm i hi, h_trans i hi⟩, subset_rfl⟩)
(fun _ hs ↦ hs.1)
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsUltraUniformity.mk_of_hasBasis
| null |
IsUltraUniformity.mem_nhds_iff_symm_trans [IsUltraUniformity X] {x : X} {s : Set X} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 X, IsSymmetricRel V ∧ IsTransitiveRel V ∧ UniformSpace.ball x V ⊆ s := by
rw [UniformSpace.mem_nhds_iff]
constructor
· rintro ⟨V, V_in, V_sub⟩
rw [IsUltraUniformity.hasBasis.mem_iff'] at V_in
obtain ⟨U, ⟨U_in, U_sym, U_trans⟩, U_sub⟩ := V_in
refine ⟨U, U_in, U_sym, U_trans, (UniformSpace.ball_mono U_sub _).trans V_sub⟩
· rintro ⟨V, V_in, _, _, V_sub⟩
exact ⟨V, V_in, V_sub⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
IsUltraUniformity.mem_nhds_iff_symm_trans
| null |
_root_.IsTransitiveRel.isOpen_ball_of_mem_uniformity (x : X) {V : Set (X × X)}
(h : IsTransitiveRel V) (h' : V ∈ 𝓤 X) :
IsOpen (ball x V) := by
rw [isOpen_iff_ball_subset]
intro y hy
exact ⟨V, h', h.ball_subset_of_mem hy⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
_root_.IsTransitiveRel.isOpen_ball_of_mem_uniformity
| null |
isClosed_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity
(x : X) {V : Set (X × X)} (h_symm : IsSymmetricRel V)
(h_trans : IsTransitiveRel V) (h' : V ∈ 𝓤 X) :
IsClosed (ball x V) := by
rw [← isOpen_compl_iff, isOpen_iff_ball_subset]
exact fun y hy ↦ ⟨V, h', fun z hyz hxz ↦ hy <| h_trans hxz <| h_symm.mk_mem_comm.mp hyz⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isClosed_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity
| null |
isClopen_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity
(x : X) {V : Set (X × X)} (h_symm : IsSymmetricRel V)
(h_trans : IsTransitiveRel V) (h' : V ∈ 𝓤 X) :
IsClopen (ball x V) :=
⟨isClosed_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity _ ‹_› ‹_› ‹_›,
h_trans.isOpen_ball_of_mem_uniformity _ ‹_›⟩
variable [IsUltraUniformity X]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
isClopen_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity
| null |
nhds_basis_clopens (x : X) :
(𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id := by
refine (nhds_basis_uniformity' (IsUltraUniformity.hasBasis)).to_hasBasis' ?_ ?_
· intro V ⟨hV, h_symm, h_trans⟩
refine ⟨ball x V, ⟨?_,
isClopen_ball_of_isSymmetricRel_of_isTransitiveRel_of_mem_uniformity _ h_symm h_trans hV⟩,
le_rfl⟩
exact mem_ball_self _ hV
· rintro u ⟨hx, hu⟩
simp [hu.right.mem_nhds_iff, hx]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
nhds_basis_clopens
| null |
_root_.TopologicalSpace.isTopologicalBasis_clopens :
TopologicalSpace.IsTopologicalBasis {s : Set X | IsClopen s} :=
.of_hasBasis_nhds fun x ↦ by simpa [and_comm] using nhds_basis_clopens x
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.Bases"
] |
Mathlib/Topology/UniformSpace/Ultra/Basic.lean
|
_root_.TopologicalSpace.isTopologicalBasis_clopens
|
A uniform space with a nonarchimedean uniformity is zero-dimensional.
|
IsUniformInducing.isUltraUniformity [IsUltraUniformity Y] {f : X → Y}
(hf : IsUniformInducing f) : IsUltraUniformity X :=
hf.comap_uniformSpace ▸ .comap inferInstance f
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsUniformInducing.isUltraUniformity
| null |
IsSymmetricRel.cauchyFilter_gen {s : Set (X × X)} (h : IsSymmetricRel s) :
IsSymmetricRel (CauchyFilter.gen s) := by
simp [IsSymmetricRel, CauchyFilter.gen, h.mem_filter_prod_comm]
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsSymmetricRel.cauchyFilter_gen
| null |
IsTransitiveRel.cauchyFilter_gen {s : Set (X × X)} (hs : IsTransitiveRel s) :
IsTransitiveRel (CauchyFilter.gen s) := by
simp only [IsTransitiveRel, CauchyFilter.gen, mem_setOf_eq]
intro f g h hfg hgh
exact hs.mem_filter_prod_comm hfg hgh
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsTransitiveRel.cauchyFilter_gen
| null |
IsUltraUniformity.cauchyFilter [IsUltraUniformity X] :
IsUltraUniformity (CauchyFilter X) := by
apply mk_of_hasBasis (CauchyFilter.basis_uniformity IsUltraUniformity.hasBasis)
· exact fun _ ⟨_, hU, _⟩ ↦ hU.cauchyFilter_gen
· exact fun _ ⟨_, _, hU⟩ ↦ hU.cauchyFilter_gen
@[simp] lemma IsUltraUniformity.cauchyFilter_iff :
IsUltraUniformity (CauchyFilter X) ↔ IsUltraUniformity X :=
⟨fun _ ↦ CauchyFilter.isUniformInducing_pureCauchy.isUltraUniformity,
fun _ ↦ inferInstance⟩
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsUltraUniformity.cauchyFilter
| null |
IsUltraUniformity.separationQuotient [IsUltraUniformity X] :
IsUltraUniformity (SeparationQuotient X) := by
have := IsUltraUniformity.hasBasis.map
(Prod.map SeparationQuotient.mk (SeparationQuotient.mk (X := X)))
rw [← SeparationQuotient.uniformity_eq] at this
apply mk_of_hasBasis this
· exact fun _ ⟨_, hU, _⟩ ↦ hU.image_prodMap _
· refine fun U ⟨hU', _, hU⟩ ↦ ?_
rintro x y z
simp only [id_eq, Set.mem_image, Prod.exists, Prod.map_apply, Prod.mk.injEq,
forall_exists_index, and_imp]
rintro a b hab rfl rfl c d hcd hc rfl
have hbc : (b, c) ∈ U := by
rw [eq_comm, SeparationQuotient.mk_eq_mk, inseparable_iff_ker_uniformity,
Filter.mem_ker] at hc
exact hc _ hU'
exact ⟨a, d, hU (hU hab hbc) hcd, by simp, by simp⟩
@[simp] lemma IsUltraUniformity.separationQuotient_iff :
IsUltraUniformity (SeparationQuotient X) ↔ IsUltraUniformity X :=
⟨fun _ ↦ SeparationQuotient.isUniformInducing_mk.isUltraUniformity,
fun _ ↦ inferInstance⟩
@[simp] lemma IsUltraUniformity.completion_iff :
IsUltraUniformity (UniformSpace.Completion X) ↔ IsUltraUniformity X := by
rw [iff_comm, ← cauchyFilter_iff, ← separationQuotient_iff]
exact Iff.rfl
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsUltraUniformity.separationQuotient
| null |
IsUltraUniformity.completion [IsUltraUniformity X] :
IsUltraUniformity (UniformSpace.Completion X) :=
completion_iff.2 inferInstance
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.UniformSpace.Ultra.Basic",
"Mathlib.Topology.UniformSpace.Ultra.Constructions"
] |
Mathlib/Topology/UniformSpace/Ultra/Completion.lean
|
IsUltraUniformity.completion
| null |
IsTransitiveRel.entourageProd {s : Set (X × X)} {t : Set (Y × Y)}
(hs : IsTransitiveRel s) (ht : IsTransitiveRel t) :
IsTransitiveRel (entourageProd s t) :=
fun _ _ _ h h' ↦ ⟨hs h.left h'.left, ht h.right h'.right⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsTransitiveRel.entourageProd
| null |
IsUltraUniformity.comap {u : UniformSpace Y} (h : IsUltraUniformity Y) (f : X → Y) :
@IsUltraUniformity _ (u.comap f) := by
letI := u.comap f
refine .mk_of_hasBasis (h.hasBasis.comap (Prod.map f f)) ?_ ?_
· exact fun _ ⟨_, hU, _⟩ ↦ hU.preimage_prodMap f
· exact fun _ ⟨_, _, hU⟩ ↦ hU.preimage_prodMap f
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.comap
| null |
IsUltraUniformity.inf {u u' : UniformSpace X} (h : @IsUltraUniformity _ u)
(h' : @IsUltraUniformity _ u') :
@IsUltraUniformity _ (u ⊓ u') := by
letI := u ⊓ u'
refine .mk_of_hasBasis (h.hasBasis.inf h'.hasBasis) ?_ ?_
· exact fun _ ⟨⟨_, hU, _⟩, _, hU', _⟩ ↦ hU.inter hU'
· exact fun _ ⟨⟨_, _, hU⟩, _, _, hU'⟩ ↦ hU.inter hU'
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.inf
| null |
IsUltraUniformity.prod [UniformSpace X] [UniformSpace Y]
[IsUltraUniformity X] [IsUltraUniformity Y] :
IsUltraUniformity (X × Y) :=
.inf (.comap ‹_› _) (.comap ‹_› _)
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.prod
|
The product of uniform spaces with nonarchimedean uniformities has a
nonarchimedean uniformity.
|
IsUltraUniformity.iInf {ι : Type*} {U : (i : ι) → UniformSpace X}
(hU : ∀ i, @IsUltraUniformity X (U i)) :
@IsUltraUniformity _ (⨅ i, U i : UniformSpace X) := by
letI : UniformSpace X := ⨅ i, U i
refine .mk_of_hasBasis (iInf_uniformity ▸ Filter.HasBasis.iInf fun i ↦ (hU i).hasBasis) ?_ ?_
· exact fun _ ⟨_, h⟩ ↦ IsSymmetricRel.iInter fun i ↦ (h i).right.left
· exact fun _ ⟨_, h⟩ ↦ IsTransitiveRel.iInter fun i ↦ (h i).right.right
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.iInf
| null |
IsUltraUniformity.pi {ι : Type*} {X : ι → Type*} [U : Π i, UniformSpace (X i)]
[h : ∀ i, IsUltraUniformity (X i)] :
IsUltraUniformity (Π i, X i) := by
suffices @IsUltraUniformity _ (⨅ i, UniformSpace.comap (Function.eval i) (U i)) by
simpa [Pi.uniformSpace_eq _] using this
exact .iInf fun i ↦ .comap (h i) (Function.eval i)
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.pi
|
The indexed product of uniform spaces with nonarchimedean uniformities has a
nonarchimedean uniformity.
|
IsUltraUniformity.bot [UniformSpace X] [DiscreteUniformity X] : IsUltraUniformity X := by
have := Filter.hasBasis_principal (idRel (α := X))
rw [← DiscreteUniformity.eq_principal_idRel] at this
apply mk_of_hasBasis this
· simpa using isSymmetricRel_idRel
· simpa using isTransitiveRel_idRel
|
instance
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.bot
| null |
IsUltraUniformity.top : @IsUltraUniformity X (⊤ : UniformSpace X) := by
letI : UniformSpace X := ⊤
have := Filter.hasBasis_top (α := (X × X))
rw [← top_uniformity] at this
apply mk_of_hasBasis this
· simpa using isSymmetricRel_univ
· simpa using isTransitiveRel_univ
|
lemma
|
Topology
|
[
"Mathlib.Topology.UniformSpace.DiscreteUniformity",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Ultra.Basic"
] |
Mathlib/Topology/UniformSpace/Ultra/Constructions.lean
|
IsUltraUniformity.top
| null |
Recurse.Config where
/-- the reducibility setting to use when comparing atoms for defeq -/
red := TransparencyMode.reducible
/-- if true, local let variables can be unfolded -/
zetaDelta := false
deriving Inhabited, BEq, Repr
attribute [nolint unusedArguments] Mathlib.Tactic.AtomM.Recurse.instReprConfig.repr
|
structure
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
Recurse.Config
|
Configuration for `AtomM.Recurse`.
|
Recurse.Context where
/-- A basically empty simp context, passed to the `simp` traversal in `AtomM.onSubexpressions`.
-/
ctx : Simp.Context
/-- A cleanup routine, which simplifies evaluation results to a more human-friendly format. -/
simp : Simp.Result → MetaM Simp.Result
|
structure
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
Recurse.Context
|
The read-only state of the `AtomM.Recurse` monad.
|
RecurseM := ReaderT Recurse.Context AtomM
|
abbrev
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
RecurseM
|
The monad for `AtomM.Recurse` contains, in addition to the `AtomM` state,
a simp context for the main traversal and a cleanup function to simplify evaluation results.
|
onSubexpressions (eval : Expr → AtomM Simp.Result) (parent : Expr) (root := true) :
RecurseM Simp.Result :=
fun nctx rctx s ↦ do
let pre : Simp.Simproc := fun e =>
try
guard <| root || parent != e -- recursion guard
let r' ← eval e rctx s
let r ← nctx.simp r'
if ← withReducible <| isDefEq r.expr e then return .done { expr := r.expr }
pure (.done r)
catch _ => pure <| .continue
let post := Simp.postDefault #[]
(·.1) <$> Simp.main parent nctx.ctx (methods := { pre, post })
|
def
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
onSubexpressions
|
A tactic in the `AtomM.RecurseM` monad which will simplify expression `parent` to a normal form, by
running a core operation `eval` (in the `AtomM` monad) on the maximal subexpression(s) on which
`eval` does not fail.
There is also a subsequent clean-up operation, governed by the context from the `AtomM.RecurseM`
monad.
* `root`: true if this is a direct call to the function.
`AtomM.RecurseM.run` sets this to `false` in recursive mode.
|
partial RecurseM.run
{α : Type} (s : IO.Ref State) (cfg : Recurse.Config) (eval : Expr → AtomM Simp.Result)
(simp : Simp.Result → MetaM Simp.Result) (x : RecurseM α) :
MetaM α := do
let ctx ← Simp.mkContext { zetaDelta := cfg.zetaDelta, singlePass := true }
(simpTheorems := #[← Elab.Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) {}])
(congrTheorems := ← getSimpCongrTheorems)
let nctx := { ctx, simp }
let rec
/-- The recursive context. -/
rctx := { red := cfg.red, evalAtom },
/-- The atom evaluator calls `AtomM.onSubexpressions` recursively. -/
evalAtom e := onSubexpressions eval e false nctx rctx s
withConfig ({ · with zetaDelta := cfg.zetaDelta }) <| x nctx rctx s
|
def
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
RecurseM.run
|
Runs a tactic in the `AtomM.RecurseM` monad, given initial data:
* `s`: a reference to the mutable `AtomM` state, for persisting across calls.
This ensures that atom ordering is used consistently.
* `cfg`: the configuration options
* `eval`: a normalization operation which will be run recursively, potentially dependent on a known
atom ordering
* `simp`: a cleanup operation which will be used to post-process expressions
* `x`: the tactic to run
|
recurse (s : IO.Ref State) (cfg : Recurse.Config)
(eval : Expr → AtomM Simp.Result)
(simp : Simp.Result → MetaM Simp.Result) (tgt : Expr) :
MetaM Simp.Result := do
RecurseM.run s cfg eval simp <| onSubexpressions eval tgt
|
def
|
Util
|
[
"Mathlib.Util.AtomM"
] |
Mathlib/Util/AtomM/Recurse.lean
|
recurse
|
Normalizes an expression, given initial data:
* `s`: a reference to the mutable `AtomM` state, for persisting across calls.
This ensures that atom ordering is used consistently.
* `cfg`: the configuration options
* `eval`: a normalization operation which will be run recursively, potentially dependent on a known
atom ordering
* `simp`: a cleanup operation which will be used to post-process expressions
* `tgt`: the expression to normalize
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.