ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : TopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Countable ↥Γ
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ⊤
meas_s_ne_zero : ¬ν s = 0
\u22a2 μ univ = ν (QuotientGroup.mk ⁻¹' univ ∩ s)
-- Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : TopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Countable ↥Γ
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ⊤
meas_s_ne_zero : ¬ν s = 0
\u22a2 μ univ ≠ ⊤
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
\u22a2 s₁.tail ⋈ s₂.tail = (s₁ ⋈ s₂).tail.tail
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
\u22a2 s₁.tail ⋈ s₂.tail = (s₁.head :: s₂.head :: (s₁.tail ⋈ s₂.tail)).tail.tail
-- Mathlib/CategoryTheory/Shift/ShiftedHom.lean
C : Type u_1
inst✝⁸ : Category.{u_5, u_1} C
D : Type u_2
inst✝⁷ : Category.{?u.33498, u_2} D
E : Type u_3
inst✝⁶ : Category.{?u.33505, u_3} E
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
inst✝² : HasShift E M
X Y Z T : C
inst✝¹ : Preadditive C
inst✝ : ∀ (a : M), (shiftFunctor C a).Additive
a b c : M
α : ShiftedHom X Y a
β₁ β₂ : ShiftedHom Y Z b
h : b + a = c
\u22a2 α.comp (β₁ + β₂) h = α.comp β₁ h + α.comp β₂ h
-- Mathlib/RingTheory/WittVector/Frobenius.lean
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n j : ℕ
hj : j < p ^ n
\u22a2 p ^ (n - v p ⟨j + 1, ⋯⟩) ∣ (p ^ n).choose (j + 1)
-- Mathlib/RingTheory/WittVector/Frobenius.lean
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n j : ℕ
hj : j < p ^ n
\u22a2 ↑(n - v p ⟨j + 1, ⋯⟩) ≤ multiplicity p ((p ^ n).choose (j + 1))
-- Mathlib/RingTheory/WittVector/Frobenius.lean
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n j : ℕ
hj : j < p ^ n
\u22a2 ↑(n - v p ⟨j + 1, ⋯⟩) ≤ ↑(n - (multiplicity p j.succ).get ⋯)
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r : R
hr : r ≠ 0
I : ↑{I | I ∈ normalizedFactors (span {r})}
\u22a2 multiplicity (↑((normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (↑I) (span {r})
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r : R
hr : r ≠ 0
I : ↑{I | I ∈ normalizedFactors (span {r})}
x : ↑{d | d ∈ normalizedFactors r}
hx : (normalizedFactorsEquivSpanNormalizedFactors hr) x = I
\u22a2 multiplicity (↑((normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (↑I) (span {r})
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r : R
hr : r ≠ 0
I : ↑{I | I ∈ normalizedFactors (span {r})}
a : R
ha : a ∈ {d | d ∈ normalizedFactors r}
hx : (normalizedFactorsEquivSpanNormalizedFactors hr) ⟨a, ha⟩ = I
\u22a2 multiplicity (↑((normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (↑I) (span {r})
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
A : C
hg : S.g = 0
inst✝ : S.HasHomology
\u22a2 S.homology ≅ S.opcycles
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
A : C
hg : S.g = 0
inst✝ : S.HasHomology
this : IsIso S.homologyι
\u22a2 S.homology ≅ S.opcycles
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
\u22a2 (W.preΨ n).coeff ((n.natAbs ^ 2 - if Even n then 4 else 1) / 2) = ↑(if Even n then n / 2 else n)
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
\u22a2 IsGδ {x}
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
U : ℕ → Set X
hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen (U i)
h_basis : (𝓝 x).HasAntitoneBasis fun i => U i
\u22a2 IsGδ {x}
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : FirstCountableTopology X
inst✝ : T1Space X
x : X
U : ℕ → Set X
hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen (U i)
h_basis : (𝓝 x).HasAntitoneBasis fun i => U i
\u22a2 IsGδ (⋂ i, ⋂ (_ : True), U i)
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
ι : Type u_5
π : ι → Type u_6
κ : Type u_7
inst✝ : TopologicalSpace X
T : (i : ι) → TopologicalSpace (π i)
f : X → (i : ι) → π i
𝔖 : Set (Set ι)
\u22a2 induced (⋃₀ 𝔖).restrict topologicalSpace = ⨅ S ∈ 𝔖, induced S.restrict topologicalSpace
-- Mathlib/Algebra/Ring/Int.lean
m n✝ : ℤ
n : ℕ
\u22a2 Odd (m ^ n) ↔ Odd m ∨ n = 0
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
x y : V
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →ₗᵢ[ℝ] F
u v : E
\u22a2 angle (f u) (f v) = angle u v
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
n : ℕ
t : Finset (Fin n)
this : ∀ (m : ℕ) (hm : n = m), (p n) (t.piecewise (fun x_1 => x) fun x => y) = (p m) ((Finset.map (finCongr hm).toEmbedding t).piecewise (fun x_1 => x) fun x => y)
\u22a2 let s := changeOriginIndexEquiv.symm ⟨n, t⟩; (((p.changeOriginSeriesTerm s.fst s.snd.fst ↑s.snd.snd ⋯) fun x_1 => x) fun x => y) = (p n) (t.piecewise (fun x_1 => x) fun x => y)
-- Mathlib/Analysis/Analytic/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
n : ℕ
t : Finset (Fin n)
this : ∀ (m : ℕ) (hm : n = m), (p n) (t.piecewise (fun x_1 => x) fun x => y) = (p m) ((Finset.map (finCongr hm).toEmbedding t).piecewise (fun x_1 => x) fun x => y)
\u22a2 (p n) (t.piecewise (fun x_1 => x) fun x => y) = (p ((changeOriginIndexEquiv.symm ⟨n, t⟩).fst + (changeOriginIndexEquiv.symm ⟨n, t⟩).snd.fst)) ((↑(changeOriginIndexEquiv.symm ⟨n, t⟩).snd.snd).piecewise (fun x_1 => x) fun x => y)
-- Mathlib/Probability/Independence/Kernel.lean
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h : ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
S : Finset ι
f : ι → Set Ω
hf : ∀ i ∈ S, f i ∈ (fun i => {s i}) i
\u22a2 ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i) = ∏ i ∈ S, (κ a) (f i)
-- Mathlib/Probability/Independence/Kernel.lean
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h : ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
S : Finset ι
f : ι → Set Ω
hf : ∀ i ∈ S, f i ∈ (fun i => {s i}) i
a : α
ha : (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
\u22a2 (κ a) (⋂ i ∈ S, f i) = ∏ i ∈ S, (κ a) (f i)
-- Mathlib/Probability/Independence/Kernel.lean
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h : ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
S : Finset ι
f : ι → Set Ω
hf : ∀ i ∈ S, f i ∈ (fun i => {s i}) i
a : α
ha : (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)
this : ∀ i ∈ S, (κ a) (f i) = (κ a) (s i)
\u22a2 (κ a) (⋂ i ∈ S, f i) = ∏ i ∈ S, (κ a) (f i)
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
n : ℕ
hn : IsPrimePow n
hn' : ¬Nat.Prime n
\u22a2 μ n = 0
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hn : IsPrimePow (p ^ k)
hn' : ¬Nat.Prime (p ^ k)
\u22a2 μ (p ^ k) = 0
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hn : IsPrimePow (p ^ k)
hn' : ¬Nat.Prime (p ^ k)
\u22a2 ¬k = 1
-- Mathlib/Data/Complex/Abs.lean
z : ℂ
\u22a2 abs z = √(z.re ^ 2 + z.im ^ 2)
-- Mathlib/Data/PNat/Interval.lean
a b : ℕ+
\u22a2 (uIcc a b).card = (↑↑b - ↑↑a).natAbs + 1
-- Mathlib/Analysis/Convolution.lean
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝²⁹ : NormedAddCommGroup E
inst✝²⁸ : NormedAddCommGroup E'
inst✝²⁷ : NormedAddCommGroup E''
inst✝²⁶ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝²⁵ : RCLike 𝕜
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : NormedSpace 𝕜 E'
inst✝²² : NormedSpace 𝕜 E''
inst✝²¹ : NormedSpace ℝ F
inst✝²⁰ : NormedSpace 𝕜 F
n : ℕ∞
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace ℝ F'
inst✝¹⁵ : NormedSpace 𝕜 F'
inst✝¹⁴ : CompleteSpace F'
inst✝¹³ : NormedAddCommGroup F''
inst✝¹² : NormedSpace ℝ F''
inst✝¹¹ : NormedSpace 𝕜 F''
inst✝¹⁰ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁹ : AddGroup G
inst✝⁸ : SFinite μ
inst✝⁷ : SFinite ν
inst✝⁶ : μ.IsAddRightInvariant
inst✝⁵ : MeasurableAdd₂ G
inst✝⁴ : MeasurableNeg G
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
hf : Integrable f ν
hg : Integrable g μ
\u22a2 ∫ (x : G), (f ⋆[L, ν] g) x ∂μ = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
-- Mathlib/Analysis/Convolution.lean
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝²⁹ : NormedAddCommGroup E
inst✝²⁸ : NormedAddCommGroup E'
inst✝²⁷ : NormedAddCommGroup E''
inst✝²⁶ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝²⁵ : RCLike 𝕜
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : NormedSpace 𝕜 E'
inst✝²² : NormedSpace 𝕜 E''
inst✝²¹ : NormedSpace ℝ F
inst✝²⁰ : NormedSpace 𝕜 F
n : ℕ∞
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace ℝ F'
inst✝¹⁵ : NormedSpace 𝕜 F'
inst✝¹⁴ : CompleteSpace F'
inst✝¹³ : NormedAddCommGroup F''
inst✝¹² : NormedSpace ℝ F''
inst✝¹¹ : NormedSpace 𝕜 F''
inst✝¹⁰ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁹ : AddGroup G
inst✝⁸ : SFinite μ
inst✝⁷ : SFinite ν
inst✝⁶ : μ.IsAddRightInvariant
inst✝⁵ : MeasurableAdd₂ G
inst✝⁴ : MeasurableNeg G
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
hf : Integrable f ν
hg : Integrable g μ
\u22a2 ∫ (y : G), ∫ (x : G), (L (f y)) (g (x - y)) ∂μ ∂ν = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
-- Mathlib/Analysis/Convolution.lean
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝²⁹ : NormedAddCommGroup E
inst✝²⁸ : NormedAddCommGroup E'
inst✝²⁷ : NormedAddCommGroup E''
inst✝²⁶ : NormedAddCommGroup F
f f' : G → E
g g' : G → E'
x x' : G
y y' : E
inst✝²⁵ : RCLike 𝕜
inst✝²⁴ : NormedSpace 𝕜 E
inst✝²³ : NormedSpace 𝕜 E'
inst✝²² : NormedSpace 𝕜 E''
inst✝²¹ : NormedSpace ℝ F
inst✝²⁰ : NormedSpace 𝕜 F
n : ℕ∞
inst✝¹⁹ : CompleteSpace F
inst✝¹⁸ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace ℝ F'
inst✝¹⁵ : NormedSpace 𝕜 F'
inst✝¹⁴ : CompleteSpace F'
inst✝¹³ : NormedAddCommGroup F''
inst✝¹² : NormedSpace ℝ F''
inst✝¹¹ : NormedSpace 𝕜 F''
inst✝¹⁰ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁹ : AddGroup G
inst✝⁸ : SFinite μ
inst✝⁷ : SFinite ν
inst✝⁶ : μ.IsAddRightInvariant
inst✝⁵ : MeasurableAdd₂ G
inst✝⁴ : MeasurableNeg G
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ E'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
hf : Integrable f ν
hg : Integrable g μ
\u22a2 ∫ (y : G), (L (f y)) (∫ (x : G), g x ∂μ) ∂ν = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : star c * c + c * star c = 2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c))
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
\u22a2 ∀ x ∈ σ ℝ a, 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
\u22a2 0 ≤ x
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : x < 0
\u22a2 False
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
\u22a2 False
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
\u22a2 0 = (-x) ^ 3
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
h_eqOn : Set.EqOn 0 (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) (σ ℝ a)
\u22a2 0 = (-x) ^ 3
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
\u22a2 a / 2 < a ↔ 0 < a
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit
\u22a2 IsPushout b.inl 0 b.snd 0
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit
\u22a2 IsPushout 0 b.inl 0 b.snd
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit
\u22a2 IsPushout (0 ≫ 0) 0 0 (b.inr ≫ b.snd)