mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : μ (⋃ n, S n) ≤ ⨆ n, μ (S n)
⊢ μ (s ∩ t) + μ (s \ t) = μ (s ∩ t) + μ (⋃ n, S n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
⊢ mapL 0 f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
⊢ mapL f 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
⊢ mapL (ContinuousLinearMap.id 𝕜 E) (ContinuousLinearMap.id 𝕜 G) = ContinuousLinearMap.id 𝕜 (E ⊗[𝕜] G)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : μ (⋃ n, S n) ≤ ⨆ n, μ (S n)
⊢ μ (s ∩ t) + μ (⋃ n, S n) ≤ μ (s ∩ t) + ⨆ n, μ (S n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ eLpNorm u (↑q) μ = eLpNorm u (↑q) (μ.restrict s)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL f g ∘SL ↑↑(commIsometry 𝕜 G E) = ↑↑(commIsometry 𝕜 H F) ∘SL mapL g f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : μ (⋃ n, S n) ≤ ⨆ n, μ (S n)
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f₁ f₂ : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL (f₁ + f₂) g = mapL f₁ g + mapL f₂ g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g₁ g₂ : G →L[𝕜] H
⊢ mapL f (g₁ + g₂) = mapL f g₁ + mapL f g₂

Tactic:
ext; simp [map_add_right]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g₁ g₂ : G →L[𝕜] H
⊢ mapL f (g₁ + g₂) = mapL f g₁ + mapL f g₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
r : 𝕜
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL (r • f) g = r • mapL f g

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: d : ℕ
⊢ μH[↑d].IsAddHaarMeasure

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
r : 𝕜
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL f (r • g) = r • mapL f g

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d : ℕ
⊢ μHE[d] = volume.addHaarScalarFactor μH[↑d] • μH[↑d]

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
basis : OrthonormalBasis (Fin 0) ℝ (EuclideanSpace ℝ (Fin 0)) := EuclideanSpace.basisFun (Fin 0) ℝ
⊢ {0} = parallelepiped ⇑basis

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ ↑(mapL f g) = map ↑f ↑g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →ₗᵢ[𝕜] F
g : G →ₗᵢ[𝕜] H
⊢ (mapIsometry f g).toContinuousLinearMap = mapL f.toContinuousLinearMap g.toContinuousLinearMap

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : InnerProductSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : InnerProductSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
A : Type u_6
B : Type u_7
inst✝³ : NormedAddCommGroup A
inst✝² : InnerProductSpace 𝕜 A
inst✝¹ : NormedAddCommGroup B
inst✝ : InnerProductSpace 𝕜 B
f₁ : E →L[𝕜] F
f₂ : A →L[𝕜] E
g₁ : G →L[𝕜] H
g₂ : B →L[𝕜] G
⊢ mapL (f₁ ∘SL f₂) (g₁ ∘SL g₂) = mapL f₁ g₁ ∘SL mapL f₂ g₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
f : E →L[𝕜] E
g : F →L[𝕜] F
n : ℕ
⊢ mapL f g ^ n = mapL (f ^ n) (g ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f₁ : E →L[𝕜] F
f₂ : A →L[𝕜] E
g : G →L[𝕜] H
⊢ mapL f₁ g ∘SL ContinuousLinearMap.rTensor G f₂ = mapL (f₁ ∘SL f₂) g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f : E →L[𝕜] F
g₁ : G →L[𝕜] H
g₂ : A →L[𝕜] G
⊢ mapL f g₁ ∘SL ContinuousLinearMap.lTensor E g₂ = mapL f (g₁ ∘SL g₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f₁ : E →L[𝕜] F
f₂ : A →L[𝕜] E
g : G →L[𝕜] H
⊢ ContinuousLinearMap.rTensor H f₁ ∘SL mapL f₂ g = mapL (f₁ ∘SL f₂) g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : ∀ (a : ℕ), ∑' (k : ℕ), μ (S (2 * k + 1 + a) \ S (2 * k + a)) ≠ ∞
⊢ ∑' (k : ℕ), μ (S (2 * k + 1) \ S (2 * k)) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : InnerProductSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace 𝕜 F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : InnerProductSpace 𝕜 G
inst✝³ : NormedAddCommGroup H
inst✝² : InnerProductSpace 𝕜 H
A : Type u_6
inst✝¹ : NormedAddCommGroup A
inst✝ : InnerProductSpace 𝕜 A
f : E →L[𝕜] F
g₁ : G →L[𝕜] H
g₂ : A →L[𝕜] G
⊢ ContinuousLinearMap.lTensor F g₁ ∘SL mapL f g₂ = mapL f (g₁ ∘SL g₂)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
f : E →L[𝕜] F
⊢ ContinuousLinearMap.rTensor G f = mapL f (ContinuousLinearMap.id 𝕜 G)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
H : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
g : G →L[𝕜] H
⊢ ContinuousLinearMap.lTensor E g = mapL (ContinuousLinearMap.id 𝕜 E) g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ ContinuousLinearMap.lTensor H f ∘SL ContinuousLinearMap.rTensor E g = mapL g f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ eLpNorm u (↑q) (μ.restrict s) ≤ eLpNorm u (↑p') (μ.restrict s) * ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹⁴ : RCLike 𝕜
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : InnerProductSpace 𝕜 E
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : InnerProductSpace 𝕜 F
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : InnerProductSpace 𝕜 G
inst✝⁷ : NormedAddCommGroup H
inst✝⁶ : InnerProductSpace 𝕜 H
inst✝⁵ : CompleteSpace E
inst✝⁴ : CompleteSpace G
inst✝³ : CompleteSpace (E ⊗[𝕜] G)
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace (F ⊗[𝕜] H)
f : E →L[𝕜] F
g : G →L[𝕜] H
⊢ ContinuousLinearMap.adjoint (mapL f g) = mapL (ContinuousLinearMap.adjoint f) (ContinuousLinearMap.adjoint g)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : InnerProductSpace 𝕜 E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : InnerProductSpace 𝕜 F
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : InnerProductSpace 𝕜 G
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace G
inst✝² : CompleteSpace (E ⊗[𝕜] G)
inst✝¹ : CompleteSpace (F ⊗[𝕜] G)
inst✝ : CompleteSpace F
f : E →L[𝕜] F
⊢ ContinuousLinearMap.adjoint (ContinuousLinearMap.rTensor G f) =
    ContinuousLinearMap.rTensor G (ContinuousLinearMap.adjoint f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
H : Type u_5
inst✝¹¹ : RCLike 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : InnerProductSpace 𝕜 E
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : InnerProductSpace 𝕜 G
inst✝⁶ : NormedAddCommGroup H
inst✝⁵ : InnerProductSpace 𝕜 H
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace G
inst✝² : CompleteSpace (E ⊗[𝕜] H)
inst✝¹ : CompleteSpace (E ⊗[𝕜] G)
inst✝ : CompleteSpace H
g : G →L[𝕜] H
⊢ ContinuousLinearMap.adjoint (ContinuousLinearMap.lTensor E g) =
    ContinuousLinearMap.lTensor E (ContinuousLinearMap.adjoint g)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝¹² : RCLike 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : InnerProductSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : InnerProductSpace 𝕜 G
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : FiniteDimensional 𝕜 E
inst✝² : FiniteDimensional 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 G
inst✝ : FiniteDimensional 𝕜 H
f : E →ₗ[𝕜] F
g : G →ₗ[𝕜] H
x✝¹ : F
x✝ : H
⊢ (adjoint (map f g)) (x✝¹ ⊗ₜ[𝕜] x✝) = (map (adjoint f) (adjoint g)) (x✝¹ ⊗ₜ[𝕜] x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ eLpNorm u (↑p') (μ.restrict s) * ↑t = eLpNorm u (↑p') μ * ↑t

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
this : ∀ (a : ℕ), ∑' (k : ℕ), μ (S (2 * k + 1 + a) \ S (2 * k + a)) ≠ ∞
⊢ ∑' (k : ℕ), μ (S (2 * k + 1 + 1) \ S (2 * k + 1)) ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: X : Type u_1
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
⊢ μHE[0] = μH[0]

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
f : E →ₗ[𝕜] F
⊢ adjoint (rTensor G f) = rTensor G (adjoint f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : InnerProductSpace 𝕜 G
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 F
inst✝ : FiniteDimensional 𝕜 G
f : E →ₗ[𝕜] F
⊢ adjoint (lTensor G f) = lTensor G (adjoint f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ HasCompactSupport u

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] (F ⊗[𝕜] G)
⊢ x = y ↔ ∀ (a : E) (b : F) (c : G), inner 𝕜 x (a ⊗ₜ[𝕜] (b ⊗ₜ[𝕜] c)) = inner 𝕜 y (a ⊗ₜ[𝕜] (b ⊗ₜ[𝕜] c))

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: d : ℕ
⊢ volume.addHaarScalarFactor μH[↑d] ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : InnerProductSpace 𝕜 G
x y : E ⊗[𝕜] (F ⊗[𝕜] G)
⊢ x = y ↔ ∀ (a : E) (b : F) (c : G), inner 𝕜 (a ⊗ₜ[𝕜] (b ⊗ₜ[𝕜] c)) x = inner 𝕜 (a ⊗ₜ[𝕜] (b ⊗ₜ[𝕜] c)) y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ eLpNorm u (↑p') μ * ↑t ≤ ↑C * eLpNorm (fderiv ℝ u) (↑p) μ * ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.TensorProduct

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
b₁ : ι₁ → E
b₂ : ι₂ → F
hb₁ : Orthonormal 𝕜 b₁
hb₂ : Orthonormal 𝕜 b₂
⊢ Orthonormal 𝕜 fun i ↦ b₁ i.1 ⊗ₜ[𝕜] b₂ i.2

Tactic:
by

Goal state after: