ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : D
f : X ⟶ Y
\u22a2 f ≫ inv ((reflectorAdjunction i).counit.app Y) = inv ((reflectorAdjunction i).counit.app X) ≫ (reflector i).map (i.map f)
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : D
f : X ⟶ Y
\u22a2 (reflectorAdjunction i).counit.app X ≫ f = (reflector i).map (i.map f) ≫ (reflectorAdjunction i).counit.app Y
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : i.EssImageSubcategory
f : X ⟶ Y
\u22a2 ((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom = (equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : i.EssImageSubcategory
f : X ⟶ Y
\u22a2 i.essImageInclusion.map (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) = i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : i.EssImageSubcategory
f : X ⟶ Y
h : (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f = (𝟭 C).map f ≫ (reflectorAdjunction i).unit.app Y.obj
\u22a2 i.essImageInclusion.map (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) = i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X Y : i.EssImageSubcategory
f : X ⟶ Y
h : (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f = f ≫ (reflectorAdjunction i).unit.app Y.obj
\u22a2 i.essImageInclusion.map (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) = i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X : D
\u22a2 i.toEssImage.map ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X) ≫ (NatIso.ofComponents equivEssImageOfReflective_counitIso_app ⋯).hom.app (i.toEssImage.obj X) = 𝟙 (i.toEssImage.obj X)
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X : D
\u22a2 i.essImageInclusion.map (i.toEssImage.map ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X) ≫ (NatIso.ofComponents equivEssImageOfReflective_counitIso_app ⋯).hom.app (i.toEssImage.obj X)) = i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))
-- Mathlib/CategoryTheory/Adjunction/Reflective.lean
C : Type u₁
D : Type u₂
E : Type u₃
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
i : D ⥤ C
inst✝ : Reflective i
X : D
\u22a2 i.essImageInclusion.map (i.toEssImage.map ((NatIso.ofComponents (fun X => (asIso ((reflectorAdjunction i).counit.app X)).symm) ⋯).hom.app X)) ≫ inv ((reflectorAdjunction i).unit.app ((𝟭 i.EssImageSubcategory).obj (i.toEssImage.obj X)).obj) = i.essImageInclusion.map (𝟙 (i.toEssImage.obj X))
-- Mathlib/Order/Interval/Finset/Nat.lean
a b c x : ℕ
\u22a2 x ∈ Ico a.succ b.succ ↔ x ∈ Ioc a b
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
\u22a2 (⋂ n, s n).Nonempty
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
\u22a2 (⋂ n, s n).Nonempty
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
this : CauchySeq u
\u22a2 (⋂ n, s n).Nonempty
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x✝ y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
this : CauchySeq u
x : α
xlim : Tendsto (fun n => u n) atTop (𝓝 x)
\u22a2 (⋂ n, s n).Nonempty
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x✝ y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
this : CauchySeq u
x : α
xlim : Tendsto (fun n => u n) atTop (𝓝 x)
n : ℕ
\u22a2 x ∈ s n
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x✝ y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
this : CauchySeq u
x : α
xlim : Tendsto (fun n => u n) atTop (𝓝 x)
n : ℕ
\u22a2 ∀ᶠ (x : ℕ) in atTop, u x ∈ s n
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
s✝ : Set α
x✝ y z : α
s : ℕ → Set α
h0 : IsComplete (s 0)
hs : ∀ (n : ℕ), IsClosed (s n)
h's : ∀ (n : ℕ), IsBounded (s n)
h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty
h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)
u : ℕ → α := fun N => ⋯.some
I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n
this : CauchySeq u
x : α
xlim : Tendsto (fun n => u n) atTop (𝓝 x)
n p : ℕ
hp : p ∈ Ici n
\u22a2 u p ∈ s n
-- Mathlib/Algebra/Order/Module/Defs.lean
α : Type u_1
β : Type u_2
a✝ a₁ a₂ : α
b b₁✝ b₂✝ : β
γ : Type u_3
inst✝⁸ : Zero α
inst✝⁷ : Preorder α
inst✝⁶ : Zero β
inst✝⁵ : Preorder β
inst✝⁴ : Zero γ
inst✝³ : Preorder γ
inst✝² : SMul α β
inst✝¹ : SMul α γ
f : β → γ
inst✝ : PosSMulReflectLE α γ
hf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂
smul : ∀ (a : α) (b : β), f (a • b) = a • f b
a : α
ha : 0 < a
b₁ b₂ : β
h : a • b₁ ≤ a • b₂
\u22a2 a • f b₁ ≤ a • f b₂
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a✝ a b : α
x✝ : ℤ
\u22a2 x✝ ∈ Int.cast ⁻¹' Icc a b ↔ x✝ ∈ Icc ⌈a⌉ ⌊b⌋
-- Mathlib/RingTheory/Smooth/Basic.lean
R S Rₘ Sₘ : Type u
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing Rₘ
inst✝¹² : CommRing Sₘ
M : Submonoid R
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : Algebra S Sₘ
inst✝⁸ : Algebra R Rₘ
inst✝⁷ : Algebra Rₘ Sₘ
inst✝⁶ : IsScalarTower R Rₘ Sₘ
inst✝⁵ : IsScalarTower R S Sₘ
inst✝⁴ : IsLocalization M Rₘ
inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Sₘ →ₐ[Rₘ] Q ⧸ I
this : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
\u22a2 ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f
-- Mathlib/RingTheory/Smooth/Basic.lean
R S Rₘ Sₘ : Type u
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing Rₘ
inst✝¹² : CommRing Sₘ
M : Submonoid R
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : Algebra S Sₘ
inst✝⁸ : Algebra R Rₘ
inst✝⁷ : Algebra Rₘ Sₘ
inst✝⁶ : IsScalarTower R Rₘ Sₘ
inst✝⁵ : IsScalarTower R S Sₘ
inst✝⁴ : IsLocalization M Rₘ
inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
\u22a2 ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f
-- Mathlib/RingTheory/Smooth/Basic.lean
R S Rₘ Sₘ : Type u
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing Rₘ
inst✝¹² : CommRing Sₘ
M : Submonoid R
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : Algebra S Sₘ
inst✝⁸ : Algebra R Rₘ
inst✝⁷ : Algebra Rₘ Sₘ
inst✝⁶ : IsScalarTower R Rₘ Sₘ
inst✝⁵ : IsScalarTower R S Sₘ
inst✝⁴ : IsLocalization M Rₘ
inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f✝ : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
f : Sₘ →ₐ[Rₘ] Q := let __src := lift I ⋯ (AlgHom.restrictScalars R f✝); { toRingHom := __src.toRingHom, commutes' := ⋯ }
\u22a2 ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f✝
-- Mathlib/RingTheory/Smooth/Basic.lean
R S Rₘ Sₘ : Type u
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing Rₘ
inst✝¹² : CommRing Sₘ
M : Submonoid R
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Algebra R Sₘ
inst✝⁹ : Algebra S Sₘ
inst✝⁸ : Algebra R Rₘ
inst✝⁷ : Algebra Rₘ Sₘ
inst✝⁶ : IsScalarTower R Rₘ Sₘ
inst✝⁵ : IsScalarTower R S Sₘ
inst✝⁴ : IsLocalization M Rₘ
inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
inst✝² : FormallySmooth R Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f✝ : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
f : Sₘ →ₐ[Rₘ] Q := let __src := lift I ⋯ (AlgHom.restrictScalars R f✝); { toRingHom := __src.toRingHom, commutes' := ⋯ }
\u22a2 (Ideal.Quotient.mkₐ Rₘ I).comp f = f✝
-- .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean
n : Nat
\u22a2 mkRat 0 n = 0
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean
α✝ : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α✝
α : Type u_3
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_4
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
inst✝ : L.NeBot
μ : Measure α
As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
\u22a2 NullMeasurableSet A μ
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean
α✝ : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α✝
α : Type u_3
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_4
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
inst✝ : L.NeBot
μ : Measure α
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
As_mble : ∀ (i : ι), AEMeasurable ((As i).indicator fun x => 1) μ
\u22a2 AEMeasurable (A.indicator fun x => 1) μ
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean
α✝ : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α✝
α : Type u_3
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_4
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
inst✝ : L.NeBot
μ : Measure α
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
As_mble : ∀ (i : ι), AEMeasurable ((As i).indicator fun x => 1) μ
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun n => (As n).indicator (fun x => 1) x) L (𝓝 (A.indicator (fun x => 1) x))
-- Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝ : SeminormedAddCommGroup β
l : Filter α
f g : α → β
hf : l.BoundedAtFilter f
hg : l.BoundedAtFilter g
\u22a2 l.BoundedAtFilter (f + g)
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
\u22a2 @splitOnP = @splitOnPTR
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
P : α → Bool
l : List α
\u22a2 splitOnP P l = splitOnPTR P l
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
P : α → Bool
l : List α
\u22a2 splitOnP P l = splitOnPTR.go P l #[] #[]
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
P : α → Bool
l : List α
\u22a2 ∀ (xs : List α) (acc : Array α) (r : Array (List α)), splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs acc.data.reverse
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
\u22a2 OpenEmbedding ⇑(colimit.ι F i).base
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
\u22a2 OpenEmbedding ⇑(colimit.ι (F ⋙ forget C) i ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
this : colimit.ι (F ⋙ forget C) i ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom = Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i
\u22a2 OpenEmbedding ⇑(colimit.ι (F ⋙ forget C) i ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
this : colimit.ι (F ⋙ forget C) i = (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
\u22a2 OpenEmbedding ⇑(colimit.ι (F ⋙ forget C) i ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
this : colimit.ι (F ⋙ forget C) i = (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
\u22a2 OpenEmbedding ⇑(((Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv) ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
this✝ : colimit.ι (F ⋙ forget C) i = (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
this : colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := i.as } ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).hom = TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) i.as
\u22a2 OpenEmbedding ⇑(((Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv) ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
i : Discrete ι
this✝ : colimit.ι (F ⋙ forget C) i = (Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
this : colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := i.as } = TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) i.as ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv
\u22a2 OpenEmbedding ⇑(((Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv) ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
as✝ : ι
this✝ : colimit.ι (F ⋙ forget C) { as := as✝ } = (Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as✝ }) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
this : colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := { as := as✝ }.as } = TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as✝ }.as ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv
\u22a2 OpenEmbedding ⇑(((Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as✝ }) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv) ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
as✝ : ι
this✝ : colimit.ι (F ⋙ forget C) { as := as✝ } = (Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as✝ }) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
this : colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := { as := as✝ }.as } = TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as✝ }.as ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv
\u22a2 OpenEmbedding ⇑((((Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as✝ }.as) ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv) ≫ (preservesColimitIso (forget C) F).inv)
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
ι : Type v
F : Discrete ι ⥤ SheafedSpace C
inst✝ : HasColimit F
as✝ : ι
this✝ : colimit.ι (F ⋙ forget C) { as := as✝ } = (Discrete.natIsoFunctor.hom.app { as := as✝ } ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := as✝ }) ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).inv
this : colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) { as := { as := as✝ }.as } = TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as✝ }.as ≫ (TopCat.sigmaIsoSigma ((F ⋙ forget C).obj ∘ Discrete.mk)).inv
\u22a2 OpenEmbedding ⇑(TopCat.sigmaι ((F ⋙ forget C).obj ∘ Discrete.mk) { as := as✝ }.as)
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
a : Ordinal.{u_1}
\u22a2 (succ a.card).ord.card.ord = (succ a.card).ord
-- Mathlib/MeasureTheory/Group/Action.lean
G : Type u
M : Type v
α : Type w
s : Set α
m : MeasurableSpace α
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace G
inst✝⁴ : MeasurableSMul G α
c : G
μ : Measure α
inst✝³ : SMulInvariantMeasure G α μ
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousConstSMul G α
inst✝ : MulAction.IsMinimal G α
K U : Set α
hK : IsCompact K
hμK : μ K ≠ 0
hU : IsOpen U
hne : U.Nonempty
t : Finset G
ht : K ⊆ ⋃ g ∈ t, g • U
hμU : μ U = 0
x✝¹ : G
x✝ : x✝¹ ∈ ↑t
\u22a2 μ (x✝¹ • U) = 0
-- Mathlib/Data/Multiset/Dedup.lean
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
s : Multiset α
\u22a2 (map f s.dedup).dedup = (map f s).dedup
-- Mathlib/Data/Nat/Choose/Multinomial.lean
α : Type u_1
s : Finset α
f✝ : α → ℕ
a b : α
n : ℕ
inst✝ : DecidableEq α
ha : a ∉ s
f : α → ℕ
\u22a2 multinomial (insert a s) f = (f a + ∑ i ∈ s, f i).choose (f a) * multinomial s f
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace X
ι : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : optParam (ι → ℝ) fun i => (μ (s i)).toReal
hsμ : autoParam ((fun i => (μ (s i)).toReal) =ᶠ[li] m) _auto✝
\u22a2 (fun i => ∫ (x : X) in s i, f x ∂μ - m i • b) =o[li] m
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace X
ι : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : optParam (ι → ℝ) fun i => (μ (s i)).toReal
hsμ : autoParam ((fun i => (μ (s i)).toReal) =ᶠ[li] m) _auto✝
\u22a2 (fun s => ∫ (x : X) in s, f x ∂μ - (μ s).toReal • b) =o[l.smallSets] fun s => (μ s).toReal
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace X
ι : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : optParam (ι → ℝ) fun i => (μ (s i)).toReal
hsμ : autoParam ((fun i => (μ (s i)).toReal) =ᶠ[li] m) _auto✝
ε : ℝ
ε₀ : 0 < ε
\u22a2 ∀ᶠ (x : Set X) in l.smallSets, ‖∫ (x : X) in x, f x ∂μ - (μ x).toReal • b‖ ≤ ε * ‖(μ x).toReal‖
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace X
ι : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s : ι → Set X
li : Filter ι
hs : Tendsto s li l.smallSets
m : optParam (ι → ℝ) fun i => (μ (s i)).toReal
hsμ : autoParam ((fun i => (μ (s i)).toReal) =ᶠ[li] m) _auto✝
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
\u22a2 ∀ᶠ (x : Set X) in l.smallSets, ‖∫ (x : X) in x, f x ∂μ - (μ x).toReal • b‖ ≤ ε * ‖(μ x).toReal‖