ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x y : A
hx : x ∈ toSubmodule S * toSubmodule T
hy : y ∈ toSubmodule S * toSubmodule T
\u22a2 x * y ∈ toSubmodule S * toSubmodule T
-- Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x y : A
hx : x ∈ toSubmodule S * toSubmodule T
hy : y ∈ toSubmodule S * toSubmodule T
this : x * y ∈ toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)
\u22a2 x * y ∈ toSubmodule S * toSubmodule T
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C : VariableChange R
\u22a2 (W.variableChange C).b₆ = ↑C.u⁻¹ ^ 6 * (W.b₆ + 2 * C.r * W.b₄ + C.r ^ 2 * W.b₂ + 4 * C.r ^ 3)
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C : VariableChange R
\u22a2 (↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) ^ 2 + 4 * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁)) = ↑C.u⁻¹ ^ 6 * (W.a₃ ^ 2 + 4 * W.a₆ + 2 * C.r * (2 * W.a₄ + W.a₁ * W.a₃) + C.r ^ 2 * (W.a₁ ^ 2 + 4 * W.a₂) + 4 * C.r ^ 3)
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝⁸ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
p : ℝ≥0∞
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : SFinite μ
f : X → ℝ≥0
s : Set X
hf : AEMeasurable f (μ.restrict s)
g : X → E
\u22a2 (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ
-- Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X : C
\u22a2 (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom
-- Mathlib/Analysis/SpecialFunctions/Exponential.lean
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕂
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
\u22a2 HasStrictDerivAt (fun u => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t
-- Mathlib/RingTheory/Presentation.lean
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
P : Presentation R S
inst✝ : P.IsFinite
\u22a2 P.ker.FG
-- Mathlib/RingTheory/Presentation.lean
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
P : Presentation R S
inst✝ : P.IsFinite
\u22a2 Ideal.span ↑⋯.toFinset = P.ker
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G : MonCat
inst✝ : MonoidalCategory V
\u22a2 (Monoidal.fromTransported (functorCategoryEquivalence V G).symm).ε.app PUnit.unit = 𝟙 ((𝟙_ (SingleObj ↑G ⥤ V)).obj PUnit.unit)
-- Mathlib/RepresentationTheory/Action/Monoidal.lean
V : Type (u + 1)
inst✝¹ : LargeCategory V
G : MonCat
inst✝ : MonoidalCategory V
\u22a2 ((functorCategoryEquivalence V G).symm.unitIso.hom.app (𝟙_ (SingleObj ↑G ⥤ V))).app PUnit.unit = 𝟙 ((𝟙_ (SingleObj ↑G ⥤ V)).obj PUnit.unit)
-- Mathlib/RingTheory/PrincipalIdealDomain.lean
R : Type u
M : Type v
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsBezout R
inst✝ : GCDMonoid R
x y : R
\u22a2 Ideal.span {gcd x y} = Ideal.span {x, y}
-- Mathlib/RingTheory/Algebraic.lean
R : Type u
S : Type u_1
A : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Ring A
inst✝³ : Algebra R A
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
a : S
x✝ : IsAlgebraic R a
f : R[X]
hf₁ : f ≠ 0
hf₂ : (aeval a) f = 0
\u22a2 (aeval ((algebraMap S A) a)) f = 0
-- Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
\u22a2 (A * C).card * B.card ≤ (A * B).card * (B / C).card
-- Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A✝ B✝ C✝ A B C : Finset α
\u22a2 (A / C⁻¹).card * B.card ≤ (A * B).card * (B * C⁻¹).card
-- Mathlib/ModelTheory/Complexity.lean
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n l : ℕ
φ✝ ψ : L.BoundedFormula α l
θ : L.BoundedFormula α l.succ
v : α → M
xs : Fin l → M
φ : L.BoundedFormula α (n + 1)
con : (∀'φ).IsAtomic
\u22a2 False
-- Mathlib/Topology/Bornology/Constructions.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝² : Bornology α
inst✝¹ : Bornology β
inst✝ : (i : ι) → Bornology (π i)
s : Set α
t : Set β
S : (i : ι) → Set (π i)
\u22a2 IsBounded (s ×ˢ s) ↔ IsBounded s
-- Mathlib/Topology/Bornology/Constructions.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝² : Bornology α
inst✝¹ : Bornology β
inst✝ : (i : ι) → Bornology (π i)
t : Set β
S : (i : ι) → Set (π i)
\u22a2 IsBounded (∅ ×ˢ ∅) ↔ IsBounded ∅
-- Mathlib/Topology/Bornology/Constructions.lean
α : Type u_1
β : Type u_2
ι : Type u_3
π : ι → Type u_4
inst✝² : Bornology α
inst✝¹ : Bornology β
inst✝ : (i : ι) → Bornology (π i)
s : Set α
t : Set β
S : (i : ι) → Set (π i)
hs : s.Nonempty
\u22a2 IsBounded (s ×ˢ s) ↔ IsBounded s
-- Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
\u22a2 volume (Metric.closedBall x r) = ENNReal.ofReal r ^ card ι * ENNReal.ofReal (√π ^ card ι / Gamma (↑(card ι) / 2 + 1))
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M✝ : Matroid α
B I✝ J X Y : Set α
e : α
M : Matroid α
hX : autoParam (X ⊆ M.E) _auto✝
I : Set α
\u22a2 I ∈ {I | M.Basis I X} ↔ I ∈ maximals (fun x x_1 => x ⊆ x_1) ({I | M.Indep I} ∩ Iic X)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a b : ℕ
ha2 : a % 2 = 1
hb2 : b % 2 = 1
\u22a2 (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a b : ℕ
ha2 : a % 2 = 1
hb2 : b % 2 = 1
ha1 : a % 4 = 1
\u22a2 (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a b : ℕ
ha2 : a % 2 = 1
hb2 : b % 2 = 1
ha3 : a % 4 = 3
\u22a2 (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a b : ℕ
ha2 : a % 2 = 1
hb2 : b % 2 = 1
ha3 : a % 4 = 3
hb1 : b % 4 = 1
\u22a2 (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a b : ℕ
ha2 : a % 2 = 1
hb2 : b % 2 = 1
ha3 : a % 4 = 3
hb3 : b % 4 = 3
\u22a2 (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
\u22a2 ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
\u22a2 ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
\u22a2 ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
\u22a2 ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
\u22a2 ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
\u22a2 ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
hle : (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm
\u22a2 ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * injectiveSeminorm x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
hle : (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm
\u22a2 ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * (fun f => ⇑f) ((normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)) x
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
hle : (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm
\u22a2 ‖((toDualContinuousMultilinearMap ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap))) x) f'‖ ≤ ‖f'‖ * ‖(toDualContinuousMultilinearMap G) x‖
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)
G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)
e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
hle : (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm
\u22a2 ‖((toDualContinuousMultilinearMap ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap))) x) f'‖ ≤ ‖(toDualContinuousMultilinearMap G) x‖ * ‖f'‖
-- Mathlib/Analysis/MellinInversion.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hf : MellinConvergent f ↑σ
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
\u22a2 mellinInv σ (mellin f) x = f x
-- Mathlib/Analysis/MellinInversion.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hFf : VerticalIntegrable (mellin f) σ volume
hfx : ContinuousAt f x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
\u22a2 mellinInv σ (mellin f) x = f x
-- Mathlib/Analysis/MellinInversion.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
hfx : ContinuousAt f x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
\u22a2 mellinInv σ (mellin f) x = f x
-- Mathlib/Analysis/MellinInversion.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
\u22a2 mellinInv σ (mellin f) x = f x
-- Mathlib/Analysis/Calculus/ContDiff/Defs.lean
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : ContDiffWithinAt 𝕜 n f s x
\u22a2 ContinuousWithinAt f s x
-- Mathlib/Analysis/Calculus/ContDiff/Defs.lean
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u✝ : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p✝ : E → FormalMultilinearSeries 𝕜 E F
h : ContDiffWithinAt 𝕜 n f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn (↑0) f p u
\u22a2 ContinuousWithinAt f s x
-- Mathlib/Analysis/Calculus/ContDiff/Defs.lean
𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u✝ : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p✝ : E → FormalMultilinearSeries 𝕜 E F
h : ContDiffWithinAt 𝕜 n f s x
u : Set E
hu : x ∈ u ∧ u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
H : HasFTaylorSeriesUpToOn (↑0) f p u
\u22a2 ContinuousWithinAt f s x