ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²³ : Semifield R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : TopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : Semifield S
inst✝¹⁷ : StarRing S
inst✝¹⁶ : MetricSpace S
inst✝¹⁵ : TopologicalSemiring S
inst✝¹⁴ : ContinuousStar S
inst✝¹³ : TopologicalSpace A
inst✝¹² : Ring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Algebra S A
inst✝⁹ : ContinuousFunctionalCalculus S q
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : CompleteSpace R
inst✝² : ContinuousFunctionalCalculus R p
inst✝¹ : UniqueContinuousFunctionalCalculus R A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
inst✝ : CompactSpace ↑(spectrum S a)
\u22a2 Continuous ⇑(starAlgHom (cfcHom hqa) h)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²³ : Semifield R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : TopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : Semifield S
inst✝¹⁷ : StarRing S
inst✝¹⁶ : MetricSpace S
inst✝¹⁵ : TopologicalSemiring S
inst✝¹⁴ : ContinuousStar S
inst✝¹³ : TopologicalSpace A
inst✝¹² : Ring A
inst✝¹¹ : StarRing A
inst✝¹⁰ : Algebra S A
inst✝⁹ : ContinuousFunctionalCalculus S q
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : CompleteSpace R
inst✝² : ContinuousFunctionalCalculus R p
inst✝¹ : UniqueContinuousFunctionalCalculus R A
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
inst✝ : CompactSpace ↑(spectrum S a)
\u22a2 (starAlgHom (cfcHom hqa) h) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝³ : FunLike F (Set α) ℝ≥0∞
inst✝² : OuterMeasureClass F α
μ : F
s✝ t : Set α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, μ (u x) = 0
\u22a2 μ s = 0
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝³ : FunLike F (Set α) ℝ≥0∞
inst✝² : OuterMeasureClass F α
μ : F
s✝ t✝ : Set α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, μ (u x) = 0
t : Set α
ht : t ⊆ s ∧ t.Countable ∧ s ⊆ ⋃ x ∈ t, u x
\u22a2 μ s = 0
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝³ : FunLike F (Set α) ℝ≥0∞
inst✝² : OuterMeasureClass F α
μ : F
s✝ t✝ : Set α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, μ (u x) = 0
t : Set α
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ ⋃ x ∈ t, u x
\u22a2 μ s = 0
-- Mathlib/MeasureTheory/OuterMeasure/Basic.lean
α : Type u_1
ι : Type u_2
F : Type u_3
inst✝³ : FunLike F (Set α) ℝ≥0∞
inst✝² : OuterMeasureClass F α
μ : F
s✝ t✝ : Set α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, μ (u x) = 0
t : Set α
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ ⋃ x ∈ t, u x
\u22a2 μ (⋃ x ∈ t, u x) = 0
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R✝ : Type u
inst✝¹ : CommRing R✝
W : Projective R✝
R : Type u
inst✝ : CommRing R
P : Fin 3 → R
u : Rˣ
\u22a2 u • P = ![↑u * P 0, ↑u * P 1, ↑u * P 2]
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R✝ : Type u
inst✝¹ : CommRing R✝
W : Projective R✝
R : Type u
inst✝ : CommRing R
P : Fin 3 → R
u : Rˣ
\u22a2 u • ![P 0, P 1, P 2] = ![↑u * ![P 0, P 1, P 2] 0, ↑u * ![P 0, P 1, P 2] 1, ↑u * ![P 0, P 1, P 2] 2]
-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
R✝ : Type u
inst✝¹ : CommRing R✝
W : Projective R✝
R : Type u
inst✝ : CommRing R
P : Fin 3 → R
u : Rˣ
\u22a2 ![u • P 0, u • P 1, u • P 2] = ![↑u * P 0, ↑u * P 1, ↑u * P 2]
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{?u.230017}
f : α → β
s : Set β
h : s ⊆ range f
\u22a2 #↑s ≤ #↑(f ⁻¹' s)
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{?u.230017}
f : α → β
s : Set β
h : s ⊆ range f
\u22a2 lift.{u, u} #↑s ≤ lift.{u, u} #↑(f ⁻¹' s)
-- Mathlib/Algebra/MvPolynomial/Equiv.lean
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e✝ : ℕ
s : σ →₀ ℕ
inst✝³ : CommSemiring R
inst✝² : CommSemiring S₁
inst✝¹ : CommSemiring S₂
inst✝ : CommSemiring S₃
e : S₁ ≃+* S₂
f : S₂ ≃+* S₃
p : MvPolynomial σ S₁
\u22a2 ((mapEquiv σ e).trans (mapEquiv σ f)) p = (mapEquiv σ (e.trans f)) p
-- Mathlib/Analysis/PSeries.lean
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)
\u22a2 ∑ k ∈ Ico (u n) (u (n + 1)), f k ≤ (u (n + 1) - u n) • f (u n)
-- Mathlib/Analysis/PSeries.lean
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)
this : ∀ k ∈ Ico (u n) (u (n + 1)), f k ≤ f (u n)
\u22a2 ∑ k ∈ Ico (u n) (u (n + 1)), f k ≤ (u (n + 1) - u n) • f (u n)
-- Mathlib/Analysis/PSeries.lean
M : Type u_1
inst✝ : OrderedAddCommMonoid M
f : ℕ → M
u : ℕ → ℕ
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : Monotone u
n : ℕ
ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)
this : ∀ k ∈ Ico (u n) (u (n + 1)), f k ≤ f (u n)
\u22a2 (u (n + 1) - u n) • f (u n) = ∑ i ∈ Ico (u n) (u (n + 1)), f (u n)
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIciTopology α
inst✝ : TopologicalSpace β
a✝ b✝ c✝ : α
f : α → β
a b c : α
H : b ∈ Ico a c
\u22a2 Iio c ∩ Ioi b ⊆ Ioo a c
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIciTopology α
inst✝ : TopologicalSpace β
a✝ b✝ c✝ : α
f : α → β
a b c : α
H : b ∈ Ico a c
\u22a2 Ioo b c ⊆ Ioo a c
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : SemilatticeInf α
a b c : α
\u22a2 Ioc a b ∩ Iic c = Ioc a (b ⊓ c)
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
\u22a2 HasSum (fun n => (↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s - ↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s) / 2) (hurwitzZetaOdd (↑a) s)
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℕ
\u22a2 (↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s - ↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s) / 2 = ↑(SignType.sign (↑↑n + a)) / ↑|↑↑n + a| ^ s / 2 + ↑(SignType.sign (↑(-(↑n + 1)) + a)) / ↑|↑(-(↑n + 1)) + a| ^ s / 2
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℕ
\u22a2 ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 + -(↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s / 2) = ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 + ↑(SignType.sign (-(↑n + 1) + a)) / ↑|-(↑n + 1) + a| ^ s / 2
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℕ
this : -(↑n + 1) + a = -(↑n + 1 - a)
\u22a2 ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 + -(↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s / 2) = ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2 + ↑(SignType.sign (-(↑n + 1) + a)) / ↑|-(↑n + 1) + a| ^ s / 2
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
\u22a2 Function.HasTemperateGrowth fun x => 0
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ
x : E
\u22a2 ‖iteratedFDeriv ℝ n (fun x => 0) x‖ ≤ 0 * (1 + ‖x‖) ^ 0
-- Mathlib/Analysis/Distribution/SchwartzSpace.lean
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ
x : E
\u22a2 0 ≤ 0 * (1 + ‖x‖) ^ 0
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a a' : α
b : β a
b' : β a'
s✝ : Finmap β
h : a ≠ a'
s : AList β
\u22a2 insert a' b' (insert a b ⟦s⟧) = insert a b (insert a' b' ⟦s⟧)
-- Mathlib/Data/Fin/Tuple/Basic.lean
m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
i : Fin (n + 1)
x : β
p : Fin n → β
\u22a2 ∀ (x_1 : Fin n), i.insertNth x p (i.succAbove x_1) = p x_1
-- Mathlib/LinearAlgebra/Dual.lean
K : Type uK
inst✝⁵ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K V₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
B : V₁ →ₗ[K] V₂ →ₗ[K] K
inst✝ : IsReflexive K V₂
\u22a2 (ker B).dualAnnihilator = range (B.dualMap ∘ₗ ↑(evalEquiv K V₂))
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : CommSemiring R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : TopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a : A
ha : autoParam (p a) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
inst✝ : UniqueContinuousFunctionalCalculus R A
f : R → R
\u22a2 cfc f 0 = (algebraMap R A) (f 0)
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
\u22a2 MeasurableSpace.pi = generateFrom (univ.pi '' univ.pi C)
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
\u22a2 MeasurableSpace.pi = generateFrom (univ.pi '' univ.pi C)
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
\u22a2 MeasurableSpace.pi ≤ generateFrom (univ.pi '' univ.pi C)
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Finite ι
inst✝ : Finite ι'
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
\u22a2 generateFrom (univ.pi '' univ.pi C) ≤ MeasurableSpace.pi
-- 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∞
f : α → ℝ≥0
\u22a2 ∑' (a : α), ↑(f a) ≠ ⊤ ↔ Summable fun a => ↑(f a)
-- 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∞
f : α → ℝ≥0
\u22a2 ∑' (a : α), ↑(f a) ≠ ⊤ ↔ Summable f
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
\u22a2 TendstoInMeasure μ f atTop g
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
\u22a2 TendstoInMeasure μ f atTop g
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
\u22a2 TendstoInMeasure μ (fun i => AEStronglyMeasurable.mk (f i) ⋯) atTop (AEStronglyMeasurable.mk g hg)
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g hg x))
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
hf_eq_ae : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), AEStronglyMeasurable.mk (f n) ⋯ x = f n x
\u22a2 ∀ᵐ (x : α) ∂μ, Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g hg x))
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
hf_eq_ae : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), AEStronglyMeasurable.mk (f n) ⋯ x = f n x
x : α
hxf : ∀ (n : ℕ), AEStronglyMeasurable.mk (f n) ⋯ x = f n x
hxg : g x = AEStronglyMeasurable.mk g hg x
hxfg : Tendsto (fun n => f n x) atTop (𝓝 (g x))
\u22a2 Tendsto (fun n => AEStronglyMeasurable.mk (f n) ⋯ x) atTop (𝓝 (AEStronglyMeasurable.mk g hg x))
-- Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
α : Type u_1
ι : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
hf_eq_ae : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), AEStronglyMeasurable.mk (f n) ⋯ x = f n x
x : α
hxf : ∀ (n : ℕ), AEStronglyMeasurable.mk (f n) ⋯ x = f n x
hxg : g x = AEStronglyMeasurable.mk g hg x
hxfg : Tendsto (fun n => f n x) atTop (𝓝 (g x))
\u22a2 Tendsto (fun n => f n x) atTop (𝓝 (g x))
-- Mathlib/Topology/Instances/Real.lean
α : Type u
β : Type v
γ : Type w
s : Set ℝ
x : ℝ
\u22a2 x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε
-- Mathlib/Dynamics/PeriodicPts.lean
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n p k : ℕ
hp : Fact (Nat.Prime p)
hk : ¬IsPeriodicPt f (p ^ k) x
hk1 : IsPeriodicPt f (p ^ (k + 1)) x
\u22a2 minimalPeriod f x = p ^ (k + 1)
-- Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
C : Type u
inst✝⁷ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁶ : HasProduct f
D : Type t
inst✝⁵ : Category.{r, t} D
inst✝⁴ : ConcreteCategory D
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j => F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b => F.obj (f b)) (forget D)
x y : (forget D).obj (F.obj (∏ᶜ f))
h : ∀ (i : J), (F.map (Pi.π f i)) x = (F.map (Pi.π f i)) y
\u22a2 x = y
-- Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
C : Type u
inst✝⁷ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁶ : HasProduct f
D : Type t
inst✝⁵ : Category.{r, t} D
inst✝⁴ : ConcreteCategory D
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j => F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b => F.obj (f b)) (forget D)
x y : (forget D).obj (F.obj (∏ᶜ f))
h : ∀ (i : J), (F.map (Pi.π f i)) x = (F.map (Pi.π f i)) y
\u22a2 (PreservesProduct.iso F f).hom x = (PreservesProduct.iso F f).hom y
-- Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
C : Type u
inst✝⁷ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁶ : HasProduct f
D : Type t
inst✝⁵ : Category.{r, t} D
inst✝⁴ : ConcreteCategory D
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j => F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b => F.obj (f b)) (forget D)
x y : (forget D).obj (F.obj (∏ᶜ f))
h : ∀ (i : J), (F.map (Pi.π f i)) x = (F.map (Pi.π f i)) y
\u22a2 ∀ (j : Discrete J), (limit.π (Discrete.functor fun b => F.obj (f b)) j) ((piComparison F f) x) = (limit.π (Discrete.functor fun b => F.obj (f b)) j) ((piComparison F f) y)
-- Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
C : Type u
inst✝⁷ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁶ : HasProduct f
D : Type t
inst✝⁵ : Category.{r, t} D
inst✝⁴ : ConcreteCategory D
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j => F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b => F.obj (f b)) (forget D)
x y : (forget D).obj (F.obj (∏ᶜ f))
h : ∀ (i : J), (F.map (Pi.π f i)) x = (F.map (Pi.π f i)) y
j : J
\u22a2 ((forget D).map (piComparison F f) ≫ (forget D).map (limit.π (Discrete.functor fun b => F.obj (f b)) { as := j })) x = ((forget D).map (piComparison F f) ≫ (forget D).map (limit.π (Discrete.functor fun b => F.obj (f b)) { as := j })) y
-- Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
C : Type u
inst✝⁷ : Category.{v, u} C
J : Type w
f : J → C
inst✝⁶ : HasProduct f
D : Type t
inst✝⁵ : Category.{r, t} D
inst✝⁴ : ConcreteCategory D
F : C ⥤ D
inst✝³ : PreservesLimit (Discrete.functor f) F
inst✝² : HasProduct fun j => F.obj (f j)
inst✝¹ : PreservesLimitsOfShape WalkingCospan (forget D)
inst✝ : PreservesLimit (Discrete.functor fun b => F.obj (f b)) (forget D)
x y : (forget D).obj (F.obj (∏ᶜ f))
h : ∀ (i : J), (F.map (Pi.π f i)) x = (F.map (Pi.π f i)) y
j : J
\u22a2 (forget D).map (F.map (Pi.π f j)) x = (forget D).map (F.map (Pi.π f j)) y
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
j : Fin n
i : Fin (n + 1)
hi : i ≠ 0
\u22a2 j ≤ i.pred hi ↔ j.succ ≤ i
-- Mathlib/Data/Finset/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : DecidableEq β
f g : α → β
s✝ : Finset α
t : Finset β
a : α
b c : β
inst✝ : DecidableEq α
s : Multiset α
x✝ : β
\u22a2 x✝ ∈ image f s.toFinset ↔ x✝ ∈ (Multiset.map f s).toFinset
-- Mathlib/MeasureTheory/Function/AEEqFun.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
\u22a2 f.pair g = mk (fun x => (↑f x, ↑g x)) ⋯
-- Mathlib/Probability/Kernel/Invariance.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
μ : Measure α
r : ℝ≥0∞
s : Set β
hs : MeasurableSet s
\u22a2 ((r • μ).bind ⇑κ) s = (r • μ.bind ⇑κ) s
-- Mathlib/Algebra/Module/PID.lean
R : Type u
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M : Type v
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type (max u v)
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
hM : Module.IsTorsion R M
\u22a2 ∃ ι x x p, ∃ (_ : ∀ (i : ι), Irreducible (p i)), ∃ e, DirectSum.IsInternal fun i => torsionBy R M (p i ^ e i)
-- Mathlib/Algebra/Module/PID.lean
R : Type u
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M : Type v
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type (max u v)
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
hM : Module.IsTorsion R M
\u22a2 Fintype { x // x ∈ (factors ⊤.annihilator).toFinset }
-- Mathlib/Algebra/Module/PID.lean
R : Type u
inst✝⁷ : CommRing R
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
M : Type v
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type (max u v)
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
hM : Module.IsTorsion R M
\u22a2 ∀ (i : { x // x ∈ (factors ⊤.annihilator).toFinset }), Irreducible (IsPrincipal.generator ↑i)
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
\u22a2 i x ∈ perfectClosure F K ↔ x ∈ perfectClosure F E
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
\u22a2 (∃ n, i x ^ ringExpChar F ^ n ∈ (algebraMap F K).range) ↔ ∃ n, x ^ ringExpChar F ^ n ∈ (algebraMap F E).range
-- Mathlib/FieldTheory/PurelyInseparable.lean
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
x✝ : ∃ n, x ^ ringExpChar F ^ n ∈ (algebraMap F E).range
n : ℕ
y : F
h : (algebraMap F E) y = x ^ ringExpChar F ^ n
\u22a2 (algebraMap F K) y = i x ^ ringExpChar F ^ n
-- Mathlib/CategoryTheory/Limits/Shapes/Images.lean
C : Type u
inst✝⁴ : Category.{v, u} C
f✝ g✝ : Arrow C
inst✝³ : HasImage f✝.hom
inst✝² : HasImage g✝.hom
sq✝ : f✝ ⟶ g✝
f g : Arrow C
inst✝¹ : HasImage f.hom
inst✝ : HasImage g.hom
sq : f ⟶ g
F G : ImageMap sq
\u22a2 F.map = G.map
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : Mul G
a₁ a₂ : G
b₁ b₂ : k
\u22a2 single a₁ b₁ * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂)
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : Mul G
a₁ a₂ : G
b₁ b₂ : k
\u22a2 (sum (single a₁ b₁) fun a₁ b₁ => sum (single a₂ b₂) fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)) = single (a₁ * a₂) (b₁ * b₂)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) (sin z)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
this : HasSum (fun i => ((-z * I) ^ i / ↑i ! - (z * I) ^ i / ↑i !) * I / 2) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
this : HasSum ((fun i => ((-z * I) ^ i / ↑i ! - (z * I) ^ i / ↑i !) * I / 2) ∘ ⇑(Nat.divModEquiv 2).symm) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
this : HasSum (fun x => ((-z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)! - (z * I) ^ (x.1 * 2 + ↑x.2) / ↑(x.1 * 2 + ↑x.2)!) * I / 2) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
z : ℂ
this : HasSum (fun x => ((-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! - (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) * I / 2) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)
\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / ↑(2 * n + 1)! / I) ((NormedSpace.exp ℂ (-z * I) - NormedSpace.exp ℂ (z * I)) * I / 2)