ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
\u22a2 ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 (g x))
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
\u22a2 ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 (g x))
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
\u22a2 ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 (g x))
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
\u22a2 ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 (g x))
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
\u22a2 ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n => (μ[g|↑ℱ n]) x) atTop (𝓝 (g x))
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
\u22a2 g =ᶠ[ae μ] limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
\u22a2 g =ᶠ[ae μ] limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 μ s < ⊤ → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 IsPiSystem {s | ∃ n, MeasurableSet s}
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 μ ∅ < ⊤ → ∫ (x : Ω) in ∅, g x ∂μ = ∫ (x : Ω) in ∅, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 ∀ t ∈ {s | ∃ n, MeasurableSet s}, μ t < ⊤ → ∫ (x : Ω) in t, g x ∂μ = ∫ (x : Ω) in t, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 ∀ (t : Set Ω), MeasurableSet t → (μ t < ⊤ → ∫ (x : Ω) in t, g x ∂μ = ∫ (x : Ω) in t, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ) → μ tᶜ < ⊤ → ∫ (x : Ω) in tᶜ, g x ∂μ = ∫ (x : Ω) in tᶜ, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
-- Mathlib/Probability/Martingale/Convergence.lean
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
R✝ : ℝ≥0
inst✝ : IsFiniteMeasure μ
g : Ω → ℝ
hg : Integrable g μ
hgmeas : StronglyMeasurable g
hle : ⨆ n, ↑ℱ n ≤ m0
hunif : UniformIntegrable (fun n => μ[g|↑ℱ n]) 1 μ
R : ℝ≥0
hR : ∀ (i : ℕ), eLpNorm ((fun n => μ[g|↑ℱ n]) i) 1 μ ≤ ↑R
hlimint : Integrable (limitProcess (fun n => μ[g|↑ℱ n]) ℱ μ) μ
this : ∀ (n : ℕ) (s : Set Ω), MeasurableSet s → ∫ (x : Ω) in s, g x ∂μ = ∫ (x : Ω) in s, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
s : Set Ω
hs : MeasurableSet s
\u22a2 ∀ (f : ℕ → Set Ω), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), μ (f i) < ⊤ → ∫ (x : Ω) in f i, g x ∂μ = ∫ (x : Ω) in f i, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ) → μ (⋃ i, f i) < ⊤ → ∫ (x : Ω) in ⋃ i, f i, g x ∂μ = ∫ (x : Ω) in ⋃ i, f i, limitProcess (fun n x => (μ[g|↑ℱ n]) x) ℱ μ x ∂μ
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ Δ' Δ'' : SimplexCategory
n : ℕ
\u22a2 Isδ₀ (SimplexCategory.δ 0)
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
\u22a2 ∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), BddBelow (range fun r => f (a, t) ↑r)
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
c : β
hc : ∀ (q : ℚ), 0 ≤ f (a, c) q
\u22a2 ∀ (q : ℚ), BddBelow (range fun r => f (a, c) ↑r)
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
c : β
hc : ∀ (q : ℚ), 0 ≤ f (a, c) q
q : ℚ
\u22a2 0 ∈ lowerBounds (range fun r => f (a, c) ↑r)
-- Mathlib/CategoryTheory/NatIso.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F G : C ⥤ D
X Y : C
α : F ⟶ G
f : X ⟶ Y
x✝ : IsIso (α.app Y)
\u22a2 α.app X ≫ G.map f ≫ inv (α.app Y) = F.map f
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑(Proj.restrict ⋯).toTopCat
\u22a2 (toSpec 𝒜 f).val.base x = (PrimeSpectrum.comap (mapId 𝒜 ⋯)) (closedPoint (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑(Proj.restrict ⋯).toTopCat
\u22a2 (PrimeSpectrum.comap (awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x)) (closedPoint ↑((Proj.restrict ⋯).presheaf.stalk x)) = (PrimeSpectrum.comap (mapId 𝒜 ⋯)) (closedPoint (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↑(Proj.restrict ⋯).toTopCat
\u22a2 ((PrimeSpectrum.comap (mapId 𝒜 ⋯)).comp (PrimeSpectrum.comap ((Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv))) (closedPoint ↑((Proj.restrict ⋯).presheaf.stalk x)) = (PrimeSpectrum.comap (mapId 𝒜 ⋯)) (closedPoint (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
-- Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
R : Type u_1
M : Type u_2
inst✝¹⁹ : CommSemiring R
inst✝¹⁸ : AddCommMonoid M
inst✝¹⁷ : Module R M
R₁ : Type u_3
M₁ : Type u_4
inst✝¹⁶ : CommRing R₁
inst✝¹⁵ : AddCommGroup M₁
inst✝¹⁴ : Module R₁ M₁
R₂ : Type u_5
M₂ : Type u_6
inst✝¹³ : CommSemiring R₂
inst✝¹² : AddCommMonoid M₂
inst✝¹¹ : Module R₂ M₂
R₃ : Type u_7
M₃ : Type u_8
inst✝¹⁰ : CommRing R₃
inst✝⁹ : AddCommGroup M₃
inst✝⁸ : Module R₃ M₃
V : Type u_9
K : Type u_10
inst✝⁷ : Field K
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module K V
B✝ : BilinForm R M
B₁ : BilinForm R₁ M₁
B₂ : BilinForm R₂ M₂
A : Type u_11
inst✝⁴ : CommRing A
inst✝³ : IsDomain A
inst✝² : Module A M₃
B₃ : BilinForm A M₃
ι : Type u_12
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
B : BilinForm A M₃
b : Basis ι A M₃
\u22a2 B.Nondegenerate ↔ ((BilinForm.toMatrix b) B).det ≠ 0
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
n : ℕ
a : α
l : Vector α n
\u22a2 Tendsto (fun p => p.1 ::ᵥ p.2) (𝓝 a ×ˢ 𝓝 l) (𝓝 (a ::ᵥ l))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
n : ℕ
a : α
l : Vector α n
\u22a2 Tendsto (fun x => ↑(x.1 ::ᵥ x.2)) (𝓝 a ×ˢ 𝓝 l) (𝓝 (a :: ↑l))
-- Mathlib/Probability/Kernel/Condexp.lean
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → F
inst✝ : TopologicalSpace F
hm : m ≤ mΩ
hf : AEStronglyMeasurable f μ
\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun ω => (id ω, id ω)) μ)
-- Mathlib/Probability/Kernel/Condexp.lean
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → F
inst✝ : TopologicalSpace F
hm : m ≤ mΩ
hf : AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun ω => (id ω, ω)) μ)
\u22a2 AEStronglyMeasurable (fun x => f x.2) (Measure.map (fun ω => (id ω, id ω)) μ)
-- Mathlib/Data/NNReal/Basic.lean
r✝ r₁ r₂ : ℝ≥0
x y : ℝ
α : Type u_1
f : α → ℝ≥0
s : Finset α
r : ℝ≥0
\u22a2 s.sup f / r = s.sup fun a => f a / r
-- Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
inst✝ : S.HasRightHomology
\u22a2 rightHomologyMap' (𝟙 S) h S.rightHomologyData ≫ S.rightHomologyData.ι = h.ι ≫ opcyclesMap' (𝟙 S) h S.rightHomologyData
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
\u22a2 (eval₂Hom f₁ g₁) p₁ = (eval₂Hom f₁ g₂) p₁
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
\u22a2 (eval₂Hom f₁ g₁) (∑ v ∈ p₁.support, (monomial v) (coeff v p₁)) = (eval₂Hom f₁ g₂) (∑ v ∈ p₁.support, (monomial v) (coeff v p₁))
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
\u22a2 (∑ x ∈ p₁.support, f₁ (coeff x p₁) * x.prod fun i k => g₁ i ^ k) = ∑ x ∈ p₁.support, f₁ (coeff x p₁) * x.prod fun i k => g₂ i ^ k
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
\u22a2 ∀ x ∈ p₁.support, (f₁ (coeff x p₁) * x.prod fun i k => g₁ i ^ k) = f₁ (coeff x p₁) * x.prod fun i k => g₂ i ^ k
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
d : σ →₀ ℕ
hd : d ∈ p₁.support
\u22a2 ∏ x ∈ d.support, g₁ x ^ d x = ∏ x ∈ d.support, g₂ x ^ d x
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
d : σ →₀ ℕ
hd : d ∈ p₁.support
\u22a2 ∀ x ∈ d.support, g₁ x ^ d x = g₂ x ^ d x
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
d : σ →₀ ℕ
hd : d ∈ p₁.support
i : σ
hi : i ∈ d.support
\u22a2 g₁ i ^ d i = g₂ i ^ d i
-- Mathlib/Algebra/MvPolynomial/Variables.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
p q : MvPolynomial σ R
inst✝ : CommSemiring S
f₁ : R →+* S
g₁ g₂ : σ → S
p₁ : MvPolynomial σ R
h : ∀ i ∈ p₁.vars, i ∈ p₁.vars → g₁ i = g₂ i
d : σ →₀ ℕ
hd : d ∈ p₁.support
i : σ
hi : i ∈ d.support
this : i ∈ p₁.vars
\u22a2 g₁ i ^ d i = g₂ i ^ d i
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal
f : ↥(simpleFunc E 1 μ)
\u22a2 ‖setToL1S T f‖ ≤ C * ‖f‖
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
T : Set α → E →L[ℝ] F
C : ℝ
hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal
f : ↥(simpleFunc E 1 μ)
\u22a2 ‖SimpleFunc.setToSimpleFunc T (toSimpleFunc f)‖ ≤ C * ∑ x ∈ (toSimpleFunc f).range, (μ (↑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
x : ℂ
\u22a2 sin (↑π / 2 - x) = cos x
-- Mathlib/Data/ENNReal/Real.lean
ι : Sort u_1
f g : ι → ℝ≥0∞
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
\u22a2 a + iInf f = ⨅ b, a + f b
-- Mathlib/Data/ENNReal/Real.lean
ι : Sort u_1
f g : ι → ℝ≥0∞
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
\u22a2 ⨅ i, f i + a = ⨅ b, a + f b
-- Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
X Y : Scheme
f : X ⟶ Y
inst✝ : IsAffine Y
\u22a2 QuasiSeparated f ↔ QuasiSeparatedSpace ↑↑X.toPresheafedSpace
-- Mathlib/Data/Nat/Factorization/Root.lean
a b n : ℕ
\u22a2 (if n = 0 ∨ a = 0 then 0 else a.factorization.prod fun p k => p ^ (k / n)) = if n = 0 ∨ a = 0 then 0 else (a.factorization ⌊/⌋ n).prod fun x x_1 => x ^ x_1
-- Mathlib/MeasureTheory/Covering/VitaliFamily.lean
α : Type u_1
inst✝ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
x : α
P : Set α → Prop
\u22a2 (∃ᶠ (a : Set α) in v.filterAt x, P a) ↔ ∀ ε > 0, ∃ a ∈ v.setsAt x, a ⊆ closedBall x ε ∧ P a
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
\u22a2 l₁ <+~ l₂ ↔ ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
\u22a2 l₁ <+~ l₂
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
this : l₁ <+~ l₂.diff l₁ ++ l₁
\u22a2 l₁ <+~ l₂
-- .lake/packages/batteries/Batteries/Data/List/Perm.lean
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
h : ∀ (x : α), x ∈ l₁ → count x l₁ ≤ count x l₂
this : l₁ <+~ l₂.diff l₁ ++ l₁
\u22a2 l₂.diff l₁ ++ l₁ ~ l₂
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : LocalRing R
h_pos : ∀ (p : ℕ), IsPrimePow p → CharP R p → P
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
\u22a2 P
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : LocalRing R
h_pos : ∀ (p : ℕ), IsPrimePow p → CharP R p → P
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
\u22a2 ∀ (p : ℕ), p ≠ 0 → CharP R p → P
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : LocalRing R
h_pos : ∀ (p : ℕ), IsPrimePow p → CharP R p → P
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
p : ℕ
p_pos : p ≠ 0
p_char : CharP R p
\u22a2 P
-- Mathlib/Algebra/CharP/MixedCharZero.lean
R : Type u_1
inst✝¹ : CommRing R
P : Prop
inst✝ : LocalRing R
h_pos : ∀ (p : ℕ), IsPrimePow p → CharP R p → P
h_equal : Algebra ℚ R → P
h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
p : ℕ
p_pos : p ≠ 0
p_char : CharP R p
p_ppow : IsPrimePow p
\u22a2 P
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
R : Type u
inst✝ : CommRing R
W : Affine R
\u22a2 evalEval 0 0 W.polynomial = -W.a₆
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁷ : OrderedRing R
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module R V
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup V'
inst✝² : Module R V'
inst✝¹ : AddTorsor V' P'
inst✝ : NoZeroSMulDivisors R V
w x z : P
h₁ : Wbtw R w x z
h : w ≠ x
h₂ : Wbtw R x w z
\u22a2 False
-- Mathlib/RepresentationTheory/Rep.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Action (Type u) (MonCat.of G)
f✝ : X ⟶ Y
A : Rep k G
f : ofMulAction k G G ⟶ A
\u22a2 (fun x => A.leftRegularHom x) ({ toFun := fun f => f.hom (Finsupp.single 1 1), map_add' := ⋯, map_smul' := ⋯ }.toFun f) = f
-- Mathlib/RepresentationTheory/Rep.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Action (Type u) (MonCat.of G)
f✝ : X ⟶ Y
A : Rep k G
f : ofMulAction k G G ⟶ A
x : G
\u22a2 (((fun x => A.leftRegularHom x) ({ toFun := fun f => f.hom (Finsupp.single 1 1), map_add' := ⋯, map_smul' := ⋯ }.toFun f)).hom ∘ₗ Finsupp.lsingle x) 1 = (f.hom ∘ₗ Finsupp.lsingle x) 1
-- Mathlib/RepresentationTheory/Rep.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Action (Type u) (MonCat.of G)
f✝ : X ⟶ Y
A : Rep k G
f : ofMulAction k G G ⟶ A
x : G
this : f.hom (((ofMulAction k G G).ρ x) (Finsupp.single 1 1)) = (A.ρ x) (f.hom (Finsupp.single 1 1))
\u22a2 (((fun x => A.leftRegularHom x) ({ toFun := fun f => f.hom (Finsupp.single 1 1), map_add' := ⋯, map_smul' := ⋯ }.toFun f)).hom ∘ₗ Finsupp.lsingle x) 1 = (f.hom ∘ₗ Finsupp.lsingle x) 1
-- Mathlib/RepresentationTheory/Rep.lean
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
X Y : Action (Type u) (MonCat.of G)
f✝ : X ⟶ Y
A : Rep k G
f : ofMulAction k G G ⟶ A
x : G
this : f.hom (((ofMulAction k G G).ρ x) (Finsupp.single 1 1)) = (A.ρ x) (f.hom (Finsupp.single 1 1))
\u22a2 f.hom (Finsupp.single x 1) = f.hom (Finsupp.single x 1)
-- Mathlib/LinearAlgebra/FreeProduct/Basic.lean
I : Type u
inst✝⁵ : DecidableEq I
i✝ : I
R : Type v
inst✝⁴ : CommSemiring R
A : I → Type w
inst✝³ : (i : I) → Semiring (A i)
inst✝² : (i : I) → Algebra R (A i)
B : Type w'
inst✝¹ : Semiring B
inst✝ : Algebra R B
maps : {i : I} → A i →ₐ[R] B
i : I
\u22a2 (ι' R A) ((lof R I A i) 1) = (mkAlgHom R A) 1
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x✝ y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Type u_4
l : Filter ι
x : ι → ℝ
hx : liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤
\u22a2 ∃ R, ∃ᶠ (n : ι) in l, R < x n
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x✝ y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Type u_4
l : Filter ι
x : ι → ℝ
hx : liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤
h : ¬∃ R, ∃ᶠ (n : ι) in l, R < x n
\u22a2 False
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x✝ y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Type u_4
l : Filter ι
x : ι → ℝ
hx : liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤
h : ∀ (x_1 : ℝ), ∀ᶠ (x_2 : ι) in l, x x_2 ≤ x_1
\u22a2 False
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r✝ p q : ℝ≥0
x✝ y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Type u_4
l : Filter ι
x : ι → ℝ
hx : liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤
h : ∀ (x_1 : ℝ), ∀ᶠ (x_2 : ι) in l, x x_2 ≤ x_1
r : ℝ≥0
\u22a2 ∀ᶠ (n : ℝ≥0∞) in map (fun n => ↑(Real.nnabs (x n))) l, ↑r ≤ n
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r✝ p q : ℝ≥0
x✝ y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Type u_4
l : Filter ι
x : ι → ℝ
hx : liminf (fun n => ↑(Real.nnabs (x n))) l ≠ ⊤
h : ∀ (x_1 : ℝ), ∀ᶠ (x_2 : ι) in l, x x_2 ≤ x_1
r : ℝ≥0
\u22a2 ∀ᶠ (a : ι) in l, r ≤ Real.nnabs (x a)