fact
stringlengths 6
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stringclasses 32
values | imports
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frontier_ball_eq_empty : frontier (ball x r) = ∅ :=
isClopen_iff_frontier_eq_empty.mp (isClopen_ball x r)
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
frontier_ball_eq_empty
| null |
closedBall_eq_or_disjoint :
closedBall x r = closedBall y r ∨ Disjoint (closedBall x r) (closedBall y r) := by
refine Set.disjoint_or_nonempty_inter (closedBall x r) (closedBall y r) |>.symm.imp
(fun h ↦ ?_) id
have h₁ := closedBall_eq_of_mem <| Set.inter_subset_left h.some_mem
have h₂ := closedBall_eq_of_mem <| Set.inter_subset_right h.some_mem
exact h₁.trans h₂.symm
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
closedBall_eq_or_disjoint
| null |
isOpen_closedBall {r : ℝ} (hr : r ≠ 0) : IsOpen (closedBall x r) := by
cases lt_or_gt_of_ne hr with
| inl h =>
simp [closedBall_eq_empty.mpr h]
| inr h =>
rw [isOpen_iff]
simp only [gt_iff_lt]
intro y hy
cases closedBall_eq_or_disjoint x y r with
| inl hd =>
use r
simp [h, hd, ball_subset_closedBall]
| inr hd =>
simp [closedBall_eq_of_mem hy, h.not_gt] at hd
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isOpen_closedBall
| null |
isClopen_closedBall {r : ℝ} (hr : r ≠ 0) : IsClopen (closedBall x r) :=
⟨Metric.isClosed_closedBall, isOpen_closedBall x hr⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isClopen_closedBall
| null |
frontier_closedBall_eq_empty {r : ℝ} (hr : r ≠ 0) : frontier (closedBall x r) = ∅ :=
isClopen_iff_frontier_eq_empty.mp (isClopen_closedBall x hr)
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
frontier_closedBall_eq_empty
| null |
isOpen_sphere {r : ℝ} (hr : r ≠ 0) : IsOpen (sphere x r) := by
rw [← closedBall_diff_ball, sdiff_eq]
exact (isOpen_closedBall x hr).inter (isClosed_ball x r).isOpen_compl
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isOpen_sphere
| null |
isClopen_sphere {r : ℝ} (hr : r ≠ 0) : IsClopen (sphere x r) :=
⟨Metric.isClosed_sphere, isOpen_sphere x hr⟩
|
lemma
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Lemmas"
] |
Mathlib/Topology/MetricSpace/Ultra/Basic.lean
|
isClopen_sphere
| null |
ContinuousMap.isUltrametricDist {X Y : Type*}
[TopologicalSpace X] [CompactSpace X] [MetricSpace Y] [IsUltrametricDist Y] :
IsUltrametricDist C(X, Y) := by
constructor
intro f g h
rw [ContinuousMap.dist_le (by positivity)]
refine fun x ↦ (dist_triangle_max (f x) (g x) (h x)).trans (max_le_max ?_ ?_) <;>
exact ContinuousMap.dist_apply_le_dist x
|
instance
|
Topology
|
[
"Mathlib.Topology.ContinuousMap.Compact",
"Mathlib.Topology.MetricSpace.Ultra.Basic"
] |
Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.lean
|
ContinuousMap.isUltrametricDist
|
Continuous maps from a compact space to an ultrametric space are an ultrametric space.
|
Pi.instIsUltrametricDist {ι : Type*} {X : ι → Type*} [Fintype ι]
[(i : ι) → PseudoMetricSpace (X i)] [(i : ι) → IsUltrametricDist (X i)] :
IsUltrametricDist ((i : ι) → X i) := by
constructor
intro f g h
simp only [dist_pi_def, ← NNReal.coe_max, NNReal.coe_le_coe, ← Finset.sup_sup]
exact Finset.sup_mono_fun fun i _ ↦ IsUltrametricDist.dist_triangle_max (f i) (g i) (h i)
|
instance
|
Topology
|
[
"Mathlib.Topology.MetricSpace.Pseudo.Pi",
"Mathlib.Topology.MetricSpace.Ultra.Basic"
] |
Mathlib/Topology/MetricSpace/Ultra/Pi.lean
|
Pi.instIsUltrametricDist
| null |
AlexDisc extends TopCat where
[is_alexandrovDiscrete : AlexandrovDiscrete carrier]
|
structure
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
AlexDisc
|
The category of Alexandrov-discrete spaces.
|
category : Category AlexDisc :=
InducedCategory.category toTopCat
|
instance
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
category
| null |
concreteCategory : ConcreteCategory AlexDisc (C(·, ·)) :=
InducedCategory.concreteCategory toTopCat
|
instance
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
concreteCategory
| null |
instHasForgetToTop : HasForget₂ AlexDisc TopCat := InducedCategory.hasForget₂ toTopCat
|
instance
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
instHasForgetToTop
| null |
forgetToTop_full : (forget₂ AlexDisc TopCat).Full where
map_surjective f := ⟨f, rfl⟩
|
instance
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
forgetToTop_full
| null |
forgetToTop_faithful : (forget₂ AlexDisc TopCat).Faithful where
|
instance
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
forgetToTop_faithful
| null |
of (X : Type*) [TopologicalSpace X] [AlexandrovDiscrete X] : AlexDisc where
toTopCat := TopCat.of X
|
abbrev
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
of
|
Construct a bundled `AlexDisc` from the underlying topological space.
|
coe_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : ↥(of α) = α := rfl
@[simp] lemma forgetToTop_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] :
(forget₂ AlexDisc TopCat).obj (of α) = TopCat.of α := rfl
@[simp] lemma coe_forgetToTop (X : AlexDisc) : ↥((forget₂ _ TopCat).obj X) = X := rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
coe_of
| null |
@[simps]
Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β where
hom := TopCat.ofHom (e : ContinuousMap α β)
inv := TopCat.ofHom (e.symm : ContinuousMap β α)
hom_inv_id := by ext; apply e.symm_apply_apply
inv_hom_id := by ext; apply e.apply_symm_apply
|
def
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
Iso.mk
|
Constructs an equivalence between preorders from an order isomorphism between them.
|
@[simps]
alexDiscEquivPreord : AlexDisc ≌ Preord where
functor := forget₂ _ _ ⋙ topToPreord
inverse.obj X := AlexDisc.of (WithUpperSet X)
inverse.map f := TopCat.ofHom (WithUpperSet.map f.hom)
unitIso := NatIso.ofComponents fun X ↦ AlexDisc.Iso.mk <| by
dsimp; exact homeoWithUpperSetTopologyorderIso X
counitIso := NatIso.ofComponents fun X ↦ Preord.Iso.mk <| by
dsimp; exact (orderIsoSpecializationWithUpperSetTopology X).symm
|
def
|
Topology
|
[
"Mathlib.Topology.Specialization"
] |
Mathlib/Topology/Order/Category/AlexDisc.lean
|
alexDiscEquivPreord
|
Sends a topological space to its specialisation order.
|
PT := FrameHom L Prop
|
abbrev
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
PT
|
The type of points of a complete lattice `L`, where a *point* of a complete lattice is,
by definition, a frame homomorphism from `L` to `Prop`.
|
@[simps]
openOfElementHom : FrameHom L (Set (PT L)) where
toFun u := {x | x u}
map_inf' a b := by simp [Set.setOf_and]
map_top' := by simp
map_sSup' S := by ext; simp [Prop.exists_iff]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
openOfElementHom
|
The frame homomorphism from a complete lattice `L` to the complete lattice of sets of
points of `L`.
|
instTopologicalSpace : TopologicalSpace (PT L) where
IsOpen s := ∃ u, {x | x u} = s
isOpen_univ := ⟨⊤, by simp⟩
isOpen_inter := by rintro s t ⟨u, rfl⟩ ⟨v, rfl⟩; use u ⊓ v; simp_rw [map_inf]; rfl
isOpen_sUnion S hS := by
choose f hf using hS
use ⨆ t, ⨆ ht, f t ht
simp_rw [map_iSup, iSup_Prop_eq, setOf_exists, hf, sUnion_eq_biUnion]
|
instance
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
instTopologicalSpace
|
The topology on the set of points of the complete lattice `L`.
|
isOpen_iff (U : Set (PT L)) : IsOpen U ↔ ∃ u : L, {x | x u} = U := Iff.rfl
|
lemma
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
isOpen_iff
|
Characterization of when a subset of the space of points is open.
|
pt : Locale ⥤ TopCat where
obj L := .of (PT L.unop)
map f := TopCat.ofHom ⟨fun p ↦ p.comp f.unop.hom,
continuous_def.2 <| by rintro s ⟨u, rfl⟩; use f.unop u; rfl⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
pt
|
The covariant functor `pt` from the category of locales to the category of
topological spaces, which sends a locale `L` to the topological space `PT L` of homomorphisms
from `L` to `Prop` and a locale homomorphism `f` to a continuous function between the spaces
of points.
|
@[simps]
localePointOfSpacePoint (x : X) : PT (Opens X) where
toFun := (x ∈ ·)
map_inf' _ _ := rfl
map_top' := rfl
map_sSup' S := by simp [Prop.exists_iff]
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
localePointOfSpacePoint
|
The unit of the adjunction between locales and topological spaces, which associates with
a point `x` of the space `X` a point of the locale of opens of `X`.
|
counitAppCont : FrameHom L (Opens <| PT L) where
toFun u := ⟨openOfElementHom L u, u, rfl⟩
map_inf' a b := by simp
map_top' := by simp
map_sSup' S := by ext; simp
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
counitAppCont
|
The counit is a frame homomorphism.
|
adjunctionTopToLocalePT : topToLocale ⊣ pt where
unit := { app := fun X ↦ TopCat.ofHom ⟨localePointOfSpacePoint X, continuous_def.2 <|
by rintro _ ⟨u, rfl⟩; simpa using u.2⟩ }
counit := { app := fun L ↦ ⟨Frm.ofHom (counitAppCont L)⟩ }
|
def
|
Topology
|
[
"Mathlib.Topology.Category.Locale"
] |
Mathlib/Topology/Order/Category/FrameAdjunction.lean
|
adjunctionTopToLocalePT
|
The forgetful functor `topToLocale` is left adjoint to the functor `pt`.
|
ContinuousOrderHom (α β : Type*) [Preorder α] [Preorder β] [TopologicalSpace α]
[TopologicalSpace β] extends OrderHom α β where
continuous_toFun : Continuous toFun
@[inherit_doc] infixr:25 " →Co " => ContinuousOrderHom
|
structure
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHom
|
The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms.
|
ContinuousOrderHomClass (F : Type*) (α β : outParam Type*) [Preorder α] [Preorder β]
[TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] : Prop
extends ContinuousMapClass F α β where
map_monotone (f : F) : Monotone f
|
class
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHomClass
|
`ContinuousOrderHomClass F α β` states that `F` is a type of continuous monotone maps.
You should extend this class when you extend `ContinuousOrderHom`.
|
@[coe]
toContinuousOrderHom (f : F) : α →Co β :=
{ toFun := f
monotone' := ContinuousOrderHomClass.map_monotone f
continuous_toFun := map_continuous f }
|
def
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
toContinuousOrderHom
|
Turn an element of a type `F` satisfying `ContinuousOrderHomClass F α β` into an actual
`ContinuousOrderHom`. This is declared as the default coercion from `F` to `α →Co β`.
|
toContinuousMap (f : α →Co β) : C(α, β) :=
{ f with }
|
def
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
toContinuousMap
|
Reinterpret a `ContinuousOrderHom` as a `ContinuousMap`.
|
instFunLike : FunLike (α →Co β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr
|
instance
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
instFunLike
| null |
@[simp] coe_toOrderHom (f : α →Co β) : ⇑f.toOrderHom = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
coe_toOrderHom
| null |
toFun_eq_coe {f : α →Co β} : f.toFun = (f : α → β) := rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
toFun_eq_coe
| null |
ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext f g h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ext
| null |
protected copy (f : α →Co β) (f' : α → β) (h : f' = f) : α →Co β :=
⟨f.toOrderHom.copy f' h, h.symm.subst f.continuous_toFun⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
copy
|
Copy of a `ContinuousOrderHom` with a new `ContinuousMap` equal to the old one. Useful to fix
definitional equalities.
|
coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
coe_copy
| null |
copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
variable (α)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
copy_eq
| null |
protected id : α →Co α :=
⟨OrderHom.id, continuous_id⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
id
|
`id` as a `ContinuousOrderHom`.
|
@[simp, norm_cast]
coe_id : ⇑(ContinuousOrderHom.id α) = id :=
rfl
variable {α}
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
coe_id
| null |
id_apply (a : α) : ContinuousOrderHom.id α a = a :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
id_apply
| null |
comp (f : β →Co γ) (g : α →Co β) : ContinuousOrderHom α γ :=
⟨f.toOrderHom.comp g.toOrderHom, f.continuous_toFun.comp g.continuous_toFun⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
comp
|
Composition of `ContinuousOrderHom`s as a `ContinuousOrderHom`.
|
coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
coe_comp
| null |
comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
comp_apply
| null |
comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) :
(f.comp g).comp h = f.comp (g.comp h) :=
rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
comp_assoc
| null |
comp_id (f : α →Co β) : f.comp (ContinuousOrderHom.id α) = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
comp_id
| null |
id_comp (f : α →Co β) : (ContinuousOrderHom.id β).comp f = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
id_comp
| null |
cancel_right {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, fun h => congr_arg₂ _ h rfl⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
cancel_right
| null |
cancel_left {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Continuous",
"Mathlib.Topology.ContinuousMap.Defs"
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
cancel_left
| null |
PseudoEpimorphism (α β : Type*) [Preorder α] [Preorder β] extends α →o β where
exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b
|
structure
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
PseudoEpimorphism
|
The type of pseudo-epimorphisms, aka p-morphisms, aka bounded maps, from `α` to `β`.
|
EsakiaHom (α β : Type*) [TopologicalSpace α] [Preorder α] [TopologicalSpace β]
[Preorder β] extends α →Co β where
exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b
|
structure
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom
|
The type of Esakia morphisms, aka continuous pseudo-epimorphisms, from `α` to `β`.
|
PseudoEpimorphismClass (F : Type*) (α β : outParam Type*)
[Preorder α] [Preorder β] [FunLike F α β] : Prop
extends RelHomClass F ((· ≤ ·) : α → α → Prop) ((· ≤ ·) : β → β → Prop) where
exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b
|
class
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
PseudoEpimorphismClass
|
`PseudoEpimorphismClass F α β` states that `F` is a type of `⊔`-preserving morphisms.
You should extend this class when you extend `PseudoEpimorphism`.
|
EsakiaHomClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α] [Preorder α]
[TopologicalSpace β] [Preorder β] [FunLike F α β] : Prop
extends ContinuousOrderHomClass F α β where
exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b
|
class
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHomClass
|
`EsakiaHomClass F α β` states that `F` is a type of lattice morphisms.
You should extend this class when you extend `EsakiaHom`.
|
instFunLike : FunLike (PseudoEpimorphism α β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr
|
instance
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
instFunLike
| null |
@[simp]
toOrderHom_eq_coe (f : PseudoEpimorphism α β) : ⇑f.toOrderHom = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
toOrderHom_eq_coe
| null |
toFun_eq_coe {f : PseudoEpimorphism α β} : f.toFun = (f : α → β) := rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
toFun_eq_coe
| null |
ext {f g : PseudoEpimorphism α β} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext f g h
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
ext
| null |
protected copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : PseudoEpimorphism α β :=
⟨f.toOrderHom.copy f' h, by simpa only [h.symm, toFun_eq_coe] using f.exists_map_eq_of_map_le'⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
copy
|
Copy of a `PseudoEpimorphism` with a new `toFun` equal to the old one. Useful to fix
definitional equalities.
|
coe_copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_copy
| null |
copy_eq (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
variable (α)
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
copy_eq
| null |
protected id : PseudoEpimorphism α α :=
⟨OrderHom.id, fun _ b h => ⟨b, h, rfl⟩⟩
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id
|
`id` as a `PseudoEpimorphism`.
|
@[simp, norm_cast]
coe_id : ⇑(PseudoEpimorphism.id α) = id := rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_id
| null |
coe_id_orderHom : (PseudoEpimorphism.id α : α →o α) = OrderHom.id := rfl
variable {α}
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_id_orderHom
| null |
id_apply (a : α) : PseudoEpimorphism.id α a = a := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id_apply
| null |
comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : PseudoEpimorphism α γ :=
⟨g.toOrderHom.comp f.toOrderHom, fun a b h₀ => by
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁
exact ⟨b, h₂, rfl⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp
|
Composition of `PseudoEpimorphism`s as a `PseudoEpimorphism`.
|
coe_comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
(g.comp f : α → γ) = g ∘ f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_comp
| null |
coe_comp_orderHom (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
(g.comp f : α →o γ) = (g : β →o γ).comp f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_comp_orderHom
| null |
comp_apply (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α) :
(g.comp f) a = g (f a) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_apply
| null |
comp_assoc (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ)
(f : PseudoEpimorphism α β) : (h.comp g).comp f = h.comp (g.comp f) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_assoc
| null |
comp_id (f : PseudoEpimorphism α β) : f.comp (PseudoEpimorphism.id α) = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_id
| null |
id_comp (f : PseudoEpimorphism α β) : (PseudoEpimorphism.id β).comp f = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id_comp
| null |
cancel_right {g₁ g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β}
(hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
cancel_right
| null |
cancel_left {g : PseudoEpimorphism β γ} {f₁ f₂ : PseudoEpimorphism α β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
cancel_left
| null |
toPseudoEpimorphism (f : EsakiaHom α β) : PseudoEpimorphism α β :=
{ f with }
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
toPseudoEpimorphism
| null |
instFunLike : FunLike (EsakiaHom α β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g
congr
|
instance
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
instFunLike
| null |
@[simp]
toContinuousOrderHom_coe {f : EsakiaHom α β} :
f.toContinuousOrderHom = (f : α → β) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
toContinuousOrderHom_coe
| null |
toFun_eq_coe {f : EsakiaHom α β} : f.toFun = (f : α → β) := rfl
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
toFun_eq_coe
| null |
ext {f g : EsakiaHom α β} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext f g h
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
ext
| null |
protected copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : EsakiaHom α β :=
⟨f.toContinuousOrderHom.copy f' h, by
simpa only [h.symm, toFun_eq_coe] using f.exists_map_eq_of_map_le'⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
copy
|
Copy of an `EsakiaHom` with a new `toFun` equal to the old one. Useful to fix definitional
equalities.
|
coe_copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_copy
| null |
copy_eq (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
variable (α)
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
copy_eq
| null |
protected id : EsakiaHom α α :=
⟨ContinuousOrderHom.id α, fun _ b h => ⟨b, h, rfl⟩⟩
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id
|
`id` as an `EsakiaHom`.
|
@[simp, norm_cast]
coe_id : ⇑(EsakiaHom.id α) = id := rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_id
| null |
coe_id_pseudoEpimorphism :
(EsakiaHom.id α : PseudoEpimorphism α α) = PseudoEpimorphism.id α := rfl
variable {α}
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_id_pseudoEpimorphism
| null |
id_apply (a : α) : EsakiaHom.id α a = a := rfl
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id_apply
| null |
coe_id_continuousOrderHom : (EsakiaHom.id α : α →Co α) = ContinuousOrderHom.id α := rfl
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_id_continuousOrderHom
| null |
comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : EsakiaHom α γ :=
⟨g.toContinuousOrderHom.comp f.toContinuousOrderHom, fun a b h₀ => by
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁
exact ⟨b, h₂, rfl⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp
|
Composition of `EsakiaHom`s as an `EsakiaHom`.
|
coe_comp_continuousOrderHom (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(g.comp f : α →Co γ) = (g : β →Co γ).comp f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_comp_continuousOrderHom
| null |
coe_comp_pseudoEpimorphism (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(g.comp f : PseudoEpimorphism α γ) = (g : PseudoEpimorphism β γ).comp f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_comp_pseudoEpimorphism
| null |
coe_comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : α → γ) = g ∘ f := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
coe_comp
| null |
comp_apply (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α) : (g.comp f) a = g (f a) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_apply
| null |
comp_assoc (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(h.comp g).comp f = h.comp (g.comp f) := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_assoc
| null |
comp_id (f : EsakiaHom α β) : f.comp (EsakiaHom.id α) = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
comp_id
| null |
id_comp (f : EsakiaHom α β) : (EsakiaHom.id β).comp f = f :=
ext fun _ => rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
id_comp
| null |
cancel_right {g₁ g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
cancel_right
| null |
cancel_left {g : EsakiaHom β γ} {f₁ f₂ : EsakiaHom α β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
|
theorem
|
Topology
|
[
"Mathlib.Order.Hom.Bounded",
"Mathlib.Topology.Order.Hom.Basic"
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
cancel_left
| null |
piOpens : C :=
∏ᶜ fun i : ι => F.obj (op (U i))
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Shapes.Equalizers",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections"
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
piOpens
|
The product of the sections of a presheaf over a family of open sets.
|
piInters : C :=
∏ᶜ fun p : ι × ι => F.obj (op (U p.1 ⊓ U p.2))
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Shapes.Equalizers",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections"
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
piInters
|
The product of the sections of a presheaf over the pairwise intersections of
a family of open sets.
|
leftRes : piOpens F U ⟶ piInters.{v'} F U :=
Pi.lift fun p : ι × ι => Pi.π _ p.1 ≫ F.map (infLELeft (U p.1) (U p.2)).op
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Shapes.Equalizers",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections"
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
leftRes
|
The morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components
are given by the restriction maps from `U i` to `U i ⊓ U j`.
|
rightRes : piOpens F U ⟶ piInters.{v'} F U :=
Pi.lift fun p : ι × ι => Pi.π _ p.2 ≫ F.map (infLERight (U p.1) (U p.2)).op
|
def
|
Topology
|
[
"Mathlib.CategoryTheory.Limits.Shapes.Equalizers",
"Mathlib.CategoryTheory.Limits.Shapes.Products",
"Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections"
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
|
rightRes
|
The morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components
are given by the restriction maps from `U j` to `U i ⊓ U j`.
|
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