mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
inst✝ : NonUnitalSeminormedRing R
s t : Set R
hs : Bornology.IsBounded s
ht : Bornology.IsBounded t
⊢ Bornology.IsBounded (s * t)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : CompleteSpace 𝕜
PE : Type u_1
PF : Type u_2
inst✝⁴ : MetricSpace PE
inst✝³ : NormedAddTorsor E PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor F PF
inst✝ : FiniteDimensional 𝕜 E
f : PE →ᵃ[𝕜] PF
⊢ ∃ K, LipschitzWith K ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → (-T) s = -1 • T s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: s t : Set ℝ≥0
hs : Bornology.IsBounded s
ht : Bornology.IsBounded t
Af : ℝ
hAf : s ⊆ closedBall 0 Af
Ag : ℝ
hAg : t ⊆ closedBall 0 Ag
key : IsCompact (closedBall 0 Af ×ˢ closedBall 0 Ag)
a✝ x : ℝ≥0
x_in_s : x ∈ s
y : ℝ≥0
y_in_t : y ∈ t
xy_eq : (fun x1 x2 ↦ x1 * x2) x y = a✝
⊢ (x, y) ∈ closedBall 0 Af ×ˢ closedBall 0 Ag ∧ (fun p ↦ p.1 * p.2) (x, y) = a✝

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
f : α → E
⊢ setToFun μ (-T) ⋯ f = -setToFun μ T hT f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: s t : Set ℝ≥0
hs : Bornology.IsBounded s
ht : Bornology.IsBounded t
⊢ Bornology.IsBounded (s * t)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hg : Integrable g μ
⊢ setToFun μ T hT (f - g) = setToFun μ T hT f - setToFun μ T hT g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ∞
hf : MemLp f p μ
A : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ
x : α
⊢ 0 x ≤ (fun a ↦ ‖f a‖ ^ p.toReal) x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
H : Type u_6
inst✝ : NormedAddCommGroup H
f : α → H
p : ℝ≥0∞
hp1 : p ≠ 0
hp2 : p ≠ ∞
hf : MemLp f p μ
⊢ eLpNorm f p μ = ENNReal.ofReal ((∫ (a : α), ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
hr : ¬c = 0
⊢ ¬Integrable (c • f) μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ f.det

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
⊢ ‖∫ (a : α), f a ∂μ‖ ≤ ∫ (a : α), ‖f a‖ ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: E' : Type u_1
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
⊢ 0 < lipschitzExtensionConstant E'

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
c : E
hμ : IsFiniteMeasure μ
⊢ (setToFun μ (weightedSMul μ) ⋯ fun a ↦ c) = μ.real univ • c

Tactic:
exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
⊢ LipschitzWith ‖↑A‖₊ ⇑A

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α → E
⊢ setToFun μ T hT (c • f) = c • setToFun μ T hT f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
c : E
⊢ ∫ (x : α), c ∂μ = μ.real univ • c

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ LipschitzWith ‖↑A.symm‖₊ ⇑A.symm

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
inst✝ : IsProbabilityMeasure μ
f : α → E
c : E
hf : ∀ᵐ (x : α) ∂μ, f x = c
⊢ ∫ (x : α), f x ∂μ = c

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → G
C : ℝ
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
⊢ ∫ (x : α), C ∂μ = C * μ.real univ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
hF : CompleteSpace F
hfi : ¬Integrable f μ
⊢ ¬Integrable g μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
hμ : Integrable f μ
hν : Integrable f ν
⊢ ∫ (x : α), f x ∂(μ + ν) = ∫ (x : α), f x ∂μ + ∫ (x : α), f x ∂ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
f : α → G
⊢ ∫ (x : α), f x ∂0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
⊢ setToFun μ T hT f = setToFun μ T hT g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h : μ = 0
⊢ f =ᵐ[μ] 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
h : μ = 0
⊢ setToFun μ T hT f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
hf : f.ker = ⊥
⊢ ∃ K > 0, AntilipschitzWith K ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
s : Finset ι
hf : ∀ i ∈ s, Integrable f (μ i)
⊢ ∫ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫ (a : α), f a ∂μ i

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ setToFun μ T hT (s.indicator fun x_1 ↦ x) = (T s) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
⊢ Injective ⇑f ↔ ∃ K > 0, AntilipschitzWith K ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
by

Goal state after: