ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
r✝ : α✝ → α✝ → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
wo : IsWellOrder α r
\u22a2 { α := ↑{o' | o' < ⟦{ α := α, r := r, wo := wo }⟧}, r := Subrel (fun x x_1 => x < x_1) {o' | o' < ⟦{ α := α, r := r, wo := wo }⟧}, wo := ⋯ } ≈ { α := ULift.{u + 1, u} { α := α, r := r, wo := wo }.α, r := ULift.down ⁻¹'o { α := α, r := r, wo := wo }.r, wo := ⋯ }
-- Mathlib/Data/ENNReal/Inv.lean
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
\u22a2 a / 3 + a / 3 + a / 3 = a
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f✝ : α → β → ℝ≥0∞
inst✝ : IsSFiniteKernel κ
f g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
\u22a2 ((κ.withDensity (f + g)) a) s✝ = ((κ.withDensity f + κ.withDensity g) a) s✝
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f✝ : α → β → ℝ≥0∞
inst✝ : IsSFiniteKernel κ
f g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
\u22a2 Measurable (g a)
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
f✝ : α → β → ℝ≥0∞
inst✝ : IsSFiniteKernel κ
f g : α → β → ℝ≥0∞
hf : Measurable (Function.uncurry f)
hg : Measurable (Function.uncurry g)
a : α
s✝ : Set β
a✝ : MeasurableSet s✝
\u22a2 Measurable (Function.uncurry (f + g))
-- Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
\u22a2 (J.sheafifyCompIso F P).inv = J.sheafifyLift (whiskerRight (J.toSheafify P) F) ⋯
-- Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
\u22a2 J.toSheafify (P ⋙ F) ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify P) F
-- Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝⁷ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝⁵ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
inst✝⁴ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F
P : Cᵒᵖ ⥤ D
inst✝³ : ConcreteCategory D
inst✝² : PreservesLimits (forget D)
inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝ : (forget D).ReflectsIsomorphisms
\u22a2 J.toSheafify (P ⋙ F) = whiskerRight (J.toSheafify P) F ≫ (J.sheafifyCompIso F P).hom
-- Mathlib/Analysis/Analytic/Constructions.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
𝕝 : Type u_7
inst✝³ : NontriviallyNormedField 𝕝
inst✝² : NormedAlgebra 𝕜 𝕝
A : Type u_8
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
e : E
f : E → F
g : E → G
w✝¹ : FormalMultilinearSeries 𝕜 E F
hf : HasFPowerSeriesAt f w✝¹ e
w✝ : FormalMultilinearSeries 𝕜 E G
hg : HasFPowerSeriesAt g w✝ e
\u22a2 AnalyticAt 𝕜 (fun x => (f x, g x)) e
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
\u22a2 IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
\u22a2 ∃ N, ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
\u22a2 padicNormE (↑f i - ↑(limSeq f i)) < ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
\u22a2 padicNormE (↑f i - ↑(limSeq f i)) < ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
\u22a2 1 ≤ (↑i + 1) * ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
\u22a2 1 ≤ ↑i * ε + 1 * ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
\u22a2 1 ≤ ↑i * ε
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : 0 < ε
N : ℕ
hN : 1 / ε < ↑N
i : ℕ
hi : i ≥ N
h : padicNormE (↑f i - ↑(Classical.choose ⋯)) < 1 / (↑i + 1)
\u22a2 0 ≤ 1 * ε
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
h : MeasurableSet s
\u22a2 (μ.restrict s).toOuterMeasure = (OuterMeasure.restrict s) μ.toOuterMeasure
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a : Ordinal.{u}
\u22a2 a.cof.ord.cof = a.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a : Ordinal.{u}
f : (b : Ordinal.{u}) → b < a.cof.ord → Ordinal.{u}
hf : a.IsFundamentalSequence a.cof.ord f
\u22a2 a.cof.ord.cof = a.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
a : Ordinal.{u}
f : (b : Ordinal.{u}) → b < a.cof.ord → Ordinal.{u}
hf : a.IsFundamentalSequence a.cof.ord f
g : (b : Ordinal.{u}) → b < a.cof.ord.cof.ord → Ordinal.{u}
hg : a.cof.ord.IsFundamentalSequence a.cof.ord.cof.ord g
\u22a2 a.cof.ord.cof = a.cof
-- Mathlib/CategoryTheory/Galois/Decomposition.lean
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
\u22a2 ∃ f, F.map f.hom b = a
-- Mathlib/CategoryTheory/Galois/Decomposition.lean
C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝² : FiberFunctor F
X A : C
inst✝¹ : IsConnected A
u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X
inst✝ : Mono u
a : ↑(F.obj A)
h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X
b : ↑(F.obj A)
\u22a2 F.map u b = F.map (CategoryTheory.PreGaloisCategory.selfProdPermIncl h b) a
-- Mathlib/Topology/NhdsSet.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Filter X
s t s₁ s₂ t₁ t₂ : Set X
x : X
\u22a2 Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t
-- Mathlib/Algebra/Group/Support.lean
α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M
\u22a2 (mulSupport f).Nonempty ↔ f ≠ 1
-- Mathlib/RingTheory/PowerBasis.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : Ring S
inst✝⁷ : Algebra R S
A : Type u_4
B : Type u_5
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : IsDomain B
inst✝³ : Algebra A B
K : Type u_6
inst✝² : Field K
S' : Type u_7
inst✝¹ : CommRing S'
inst✝ : Algebra R S'
pb : PowerBasis R S
e : S ≃ₐ[R] S'
i : Fin pb.dim
\u22a2 (pb.basis.map e.toLinearEquiv) i = e pb.gen ^ ↑i
-- Mathlib/NumberTheory/Divisors.lean
n✝ : ℕ
α : Type u_1
inst✝ : CommMonoid α
n : ℕ
f : ℕ → α
hn : ¬n = 0
\u22a2 ∏ d ∈ n.divisors, f (n / d) = n.divisors.prod f
-- Mathlib/NumberTheory/Divisors.lean
n✝ : ℕ
α : Type u_1
inst✝ : CommMonoid α
n : ℕ
f : ℕ → α
hn : ¬n = 0
\u22a2 ∏ x ∈ image (HDiv.hDiv n) n.divisors, f x = n.divisors.prod f
-- Mathlib/NumberTheory/Divisors.lean
n✝ : ℕ
α : Type u_1
inst✝ : CommMonoid α
n : ℕ
f : ℕ → α
hn : ¬n = 0
\u22a2 ∀ x ∈ n.divisors, ∀ y ∈ n.divisors, n / x = n / y → x = y
-- Mathlib/Geometry/Manifold/InteriorBoundary.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : SmoothManifoldWithCorners J N
x : M
y : N
\u22a2 (I.prod J).boundary (M × N) = univ.prod (J.boundary N) ∪ (I.boundary M).prod univ
-- Mathlib/Geometry/Manifold/InteriorBoundary.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : SmoothManifoldWithCorners J N
x : M
y : N
h : (I.prod J).boundary (M × N) = (I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ := Trans.trans (Trans.trans (Trans.trans rfl (boundary_eq_complement_interior (I.prod J))) (Eq.mpr (id (congrArg (fun _a => _aᶜ = (I.interior M ×ˢ J.interior N)ᶜ) (interior_prod J))) (Eq.refl (I.interior M ×ˢ J.interior N)ᶜ))) (Eq.mpr (id (congrArg (fun _a => _a = (I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ) (compl_prod_eq_union (I.interior M) (J.interior N)))) (Eq.refl ((I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ)))
\u22a2 (I.prod J).boundary (M × N) = univ.prod (J.boundary N) ∪ (I.boundary M).prod univ
-- Mathlib/Geometry/Manifold/InteriorBoundary.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
J : ModelWithCorners 𝕜 E' H'
inst✝ : SmoothManifoldWithCorners J N
x : M
y : N
h : (I.prod J).boundary (M × N) = (I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ := Trans.trans (Trans.trans (Trans.trans rfl (boundary_eq_complement_interior (I.prod J))) (Eq.mpr (id (congrArg (fun _a => _aᶜ = (I.interior M ×ˢ J.interior N)ᶜ) (interior_prod J))) (Eq.refl (I.interior M ×ˢ J.interior N)ᶜ))) (Eq.mpr (id (congrArg (fun _a => _a = (I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ) (compl_prod_eq_union (I.interior M) (J.interior N)))) (Eq.refl ((I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ)))
\u22a2 univ ×ˢ (J.interior N)ᶜ ∪ (I.interior M)ᶜ ×ˢ univ = univ.prod (J.interior N)ᶜ ∪ (I.interior M)ᶜ.prod univ
-- Mathlib/Algebra/Order/Interval/Set/Group.lean
α : Type u_1
inst✝ : OrderedRing α
a : α
\u22a2 Pairwise (Disjoint on fun n => Ioc (↑n) (↑n + 1))
-- Mathlib/RingTheory/WittVector/Defs.lean
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
z : ℤ
n : ℕ
\u22a2 constantCoeff (wittZSMul p z n) = 0
-- Mathlib/RingTheory/WittVector/Defs.lean
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
z : ℤ
n : ℕ
\u22a2 constantCoeff (z • X 0) = 0
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t✝ : Set α
μ : ι → Measure α
s : Set α
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
\u22a2 ((sum μ).restrict s) t = (sum fun i => (μ i).restrict s) t
-- Mathlib/Order/Filter/EventuallyConst.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f : α → β
p : α → Prop
\u22a2 EventuallyConst p l ↔ (∀ᶠ (x : α) in l, p x) ∨ ∀ᶠ (x : α) in l, ¬p x
-- Mathlib/Data/NNReal/Basic.lean
r r₁ r₂ : ℝ≥0
x y : ℝ
\u22a2 1 < ↑r ↔ 1 < r
-- Mathlib/Topology/ContinuousFunction/Bounded.lean
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f g : α →ᵇ β
x : α
C : ℝ
\u22a2 ⇑(nsmulRec 0 f) = 0 • ⇑f
-- Mathlib/Topology/ContinuousFunction/Bounded.lean
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f g : α →ᵇ β
x : α
C : ℝ
n : ℕ
\u22a2 ⇑(nsmulRec (n + 1) f) = (n + 1) • ⇑f
-- Mathlib/GroupTheory/Coprod/Basic.lean
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
\u22a2 inl.range ⊔ inr.range = ⊤
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : LocallyRingedSpace
f : X ⟶ Y
H : IsOpenImmersion f
\u22a2 SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom ≫ (Y.ofRestrict ⋯).val = f.val
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : LocallyRingedSpace
f : X ⟶ Y
H : IsOpenImmersion f
\u22a2 SheafedSpace.forgetToPresheafedSpace.map (SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom ≫ (Y.ofRestrict ⋯).val) = SheafedSpace.forgetToPresheafedSpace.map f.val
-- Mathlib/Geometry/RingedSpace/OpenImmersion.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : LocallyRingedSpace
f : X ⟶ Y
H : IsOpenImmersion f
\u22a2 (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom ≫ SheafedSpace.forgetToPresheafedSpace.map (Y.ofRestrict ⋯).val = SheafedSpace.forgetToPresheafedSpace.map f.val
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
hf : (mulSupport f).Finite
\u22a2 ∏ᶠ (i : α), f i = ∏ i ∈ hf.toFinset, f i
-- Mathlib/GroupTheory/Index.lean
G : Type u_1
G' : Type u_2
inst✝¹ : Group G
inst✝ : Group G'
H K L : Subgroup G
\u22a2 H.relindex (K ⊓ L) * K.relindex L = (H ⊓ K).relindex L
-- Mathlib/Topology/LocallyConstant/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
x✝ : LocallyConstant X Z
\u22a2 comap e.toContinuousMap (comap e.symm.toContinuousMap x✝) = x✝
-- Mathlib/Topology/LocallyConstant/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : X ≃ₜ Y
x✝ : LocallyConstant Y Z
\u22a2 comap e.symm.toContinuousMap (comap e.toContinuousMap x✝) = x✝
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
l₁ l₂ : List α
a : α
s : l₁ ⊆ l₂
x✝ : α
\u22a2 x✝ ∈ a :: l₁ → x✝ ∈ a :: l₂
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
l₁ l₂ : List α
a : α
s : l₁ ⊆ l₂
x✝ : α
\u22a2 x✝ = a ∨ x✝ ∈ l₁ → x✝ = a ∨ x✝ ∈ l₂
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 f (limsup (fun n => (⇑f)^[n] a) atTop) = limsup (fun n => (⇑f)^[n] a) atTop
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ⨅ i, f (⨆ i_1, (⇑f)^[i_1 + i] a) = ⨅ n, ⨆ i, (⇑f)^[i + n] a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ⨅ i, ⨆ i_1, (⇑f)^[i_1 + i.succ] a = ⨅ n, ⨆ i, (⇑f)^[i + n] a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ⨅ i, ⨆ i_1, (⇑f)^[i_1 + i.succ] a = (⨅ i, ⨅ (_ : 0 < i), ⨆ i_1, (⇑f)^[i_1 + i] a) ⊓ ⨅ i, ⨅ (_ : ¬0 < i), ⨆ i_1, (⇑f)^[i_1 + i] a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ⨅ i, ⨆ i_1, (⇑f)^[i_1 + i.succ] a ≤ ⨆ i, (⇑f)^[i] a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ⨆ j, (⇑f)^[j + (0 + 1)] a ≤ ⨆ i, (⇑f)^[i] a
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝ : CompleteLattice α
f : CompleteLatticeHom α α
a : α
\u22a2 ∀ (i : ℕ), (⇑f)^[i + 1] a ≤ ⨆ i, (⇑f)^[i] a
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : IsEmpty α
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
this : {b | BddAbove (range fun i => f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
\u22a2 MeasurableSet {b | BddAbove (range fun i => f i b)}
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Sort y
s t u✝ : Set α
inst✝¹⁰ : TopologicalSpace α
inst✝⁹ : MeasurableSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : TopologicalSpace β
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
this : {b | BddAbove (range fun i => f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
\u22a2 MeasurableSet (⋃ i, {x | ∀ (i_1 : ι), f i_1 x ≤ u i})