mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case pos
X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁶ : MeasurableSpace X
inst✝⁵ : CommMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : PseudoMetrizableSpace E
inst✝² : ContinuousMul E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hm : Multipliable (fun i ↦ f i) L
h✝¹ : L.HasSupport ∧ ((Function.mulSupport fun i ↦ f i) ∩ L.support).Finite
h✝ : Function.HasFiniteMulSupport ((L.support.mulIndicator fun i ↦ f i) ∘ PLift.down)
⊢ AEStronglyMeasurable (∏ i ∈ Finite.toFinset h✝, L.support.mulIndicator (fun i ↦ f i) i.down) μ

Tactic:
refine Finset.aestronglyMeasurable_prod _ (fun _ _ ↦ ?_)
    rw [Set.mulIndicator]
    split_ifs <;> fun_prop

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case refine_2
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
⊢ ∀ (x : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')), ‖(flipMultilinearEquivₗ 𝕜 E G G').symm x‖ ≤ 1 * ‖x‖

Tactic:
intro f
    suffices ‖f.flipLinear‖ ≤ ‖f‖ by simpa
    apply MultilinearMap.mkContinuousLinear_norm_le
    positivity

Goal state after:
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace ε
inst✝³ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝² : PseudoMetrizableSpace ε
a : X
inst✝¹ : LinearOrder X
inst✝ : CompactIccSpace X
⊢ IntegrableOn f (Iic a) μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn f (Iic a) μ

Tactic:
refine ⟨fun h ↦ ⟨⟨Iic a, Iic_mem_atBot a, h⟩, h.locallyIntegrableOn⟩, fun ⟨⟨s, hsl, hs⟩, h⟩ ↦ ?_⟩

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace ε
inst✝³ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝² : PseudoMetrizableSpace ε
a : X
inst✝¹ : LinearOrder X
inst✝ : CompactIccSpace X
x✝ : IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn f (Iic a) μ
s : Set X
hsl : s ∈ atBot
hs : IntegrableOn f s μ
h : LocallyIntegrableOn f (Iic a) μ
⊢ IntegrableOn f (Iic a) μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Bornology.IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
A : MemLp (fun x ↦ C) ∞ (μ.restrict s)
⊢ ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ ‖C‖

Tactic:
apply (ae_restrict_iff' h's).mpr

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : Bornology.IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
A : MemLp (fun x ↦ C) ∞ (μ.restrict s)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ ‖C‖
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case H
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
y : Eₗ
x : E
hxy : e x = y
⊢ (g ∘SL e.range.subtypeL) ⟨y, ⋯⟩ = (f.compLeftInverse e) ⟨y, ⋯⟩

Tactic:
compLeftInverse_apply_of_bdd _ _ ⟨C, h_norm⟩ x y hxy

Goal state after:
case H
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
y : Eₗ
x : E
hxy : e x = y
⊢ (g ∘SL e.range.subtypeL) ⟨y, ⋯⟩ = f x
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case H
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
y : Eₗ
x : E
hxy : e x = y
⊢ (g ∘SL e.range.subtypeL) ⟨y, ⋯⟩ = f x

Tactic:
simp [← hxy, ← H]

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

Goal state before:
pp: case inr
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ f.range
⊢ (T (⇑f ⁻¹' {x})) x = 0

Tactic:
SimpleFunc.mem_range

Goal state after:
case inr
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ Set.range ⇑f
⊢ (T (⇑f ⁻¹' {x})) x = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case inr
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ f.range
⊢ μ.real (⇑f ⁻¹' {x}) • x = 0

Tactic:
SimpleFunc.mem_range

Goal state after:
case inr
α : Type u_1
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : DecidablePred fun x ↦ x ≠ 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
x : F
hx✝ : x ∉ f.range ∨ x = 0
hx : x ∉ Set.range ⇑f
⊢ μ.real (⇑f ⁻¹' {x}) • x = 0
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

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
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
h : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
⊢ z ∈ ⇑f ⁻¹' {x} ∧ z ∈ ⇑g ⁻¹' {y} → z ∈ {a | ¬f a = g a}

Tactic:
Set.mem_ofPred_eq,

Goal state after:
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
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
h : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
⊢ z ∈ ⇑f ⁻¹' {x} ∧ z ∈ ⇑g ⁻¹' {y} → ¬f z = g z
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range with x ≠ 0, (T' (⇑f ⁻¹' {x})) x = c • ∑ x ∈ f.range with x ≠ 0, (T (⇑f ⁻¹' {x})) x

Tactic:
suffices ∀ x ∈ {x ∈ f.range | x ≠ 0}, T' (f ⁻¹' {x}) = c • T (f ⁻¹' {x}) by
    rw [smul_sum]
    refine Finset.sum_congr rfl fun x hx => ?_
    rw [this x hx, _root_.smul_apply]

Goal state after:
α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F'
c : ℝ
h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T' s = c • T s
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∀ x ∈ {x ∈ f.range | x ≠ 0}, T' (⇑f ⁻¹' {x}) = c • T (⇑f ⁻¹' {x})
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 g : α → G
hf : Integrable f μ
hg : Integrable g μ
⊢ edist (∫ (a : α), f a ∂μ) (∫ (a : α), g a ∂μ) ≤ ∫⁻ (a : α), edist (f a) (g a) ∂μ

Tactic:
rw [edist_dist]
  exact ENNReal.ofReal_le_of_le_toReal (dist_integral_le_lintegral_edist hf hg)

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
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)

Tactic:
simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
hF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
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
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)
s : Set α
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)

Tactic:
simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
s : Set α
hF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ ∫⁻ (x : α), ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
huv : Tendsto (u / v) l (𝓝 1)
this : ∀ᶠ (x : α) in l, v x = 0 → u x = 0
⊢ u ~[l] v

Tactic:
rw [isEquivalent_iff_exists_eq_mul]
    exact ⟨u / v, huv, this.mono fun x hz' ↦ (div_mul_cancel_of_imp hz').symm⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
φ : α → 𝕜
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ C * ‖v x‖
c : ℝ
huv : ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
hφ : ∀ᶠ (x : α) in l, ‖φ x - 1‖ < c / 2 / C
hc : 0 < c
x : α
hCuvx : ‖u x‖ ≤ C * ‖v x‖
huvx : ‖(u - v) x‖ ≤ c / 2 * ‖v x‖
hφx : ‖φ x - 1‖ < c / 2 / C
key : ‖φ x - 1‖ * ‖u x‖ ≤ c / 2 * ‖v x‖
⊢ ‖φ x • u x - v x‖ ≤ c * ‖v x‖

Tactic:
calc
    ‖((fun x : α ↦ φ x • u x) - v) x‖ = ‖(φ x - 1) • u x + (u x - v x)‖ := by
      simp [sub_smul, sub_add]
    _ ≤ ‖(φ x - 1) • u x‖ + ‖u x - v x‖ := norm_add_le _ _
    _ = ‖φ x - 1‖ * ‖u x‖ + ‖u x - v x‖ := by rw [norm_smul]
    _ ≤ c / 2 * ‖v x‖ + ‖u x - v x‖ := by gcongr
    _ ≤ c / 2 * ‖v x‖ + c / 2 * ‖v x‖ := by gcongr; exact huvx
    _ = c * ‖v x‖ := by ring

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : ClosedIciTopology E
f g : α → E
hf : 0 ≤ᵐ[μ] f
hgi : Integrable g μ
h : f ≤ᵐ[μ] g
hfi : ¬Integrable f μ
⊢ ∫ (a : α), f a ∂μ ≤ ∫ (a : α), g a ∂μ

Tactic:
exact integral_undef hfi ▸ integral_nonneg_of_ae (hf.trans h)

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h : x ≠ 0
⊢ ‖↑‖x‖ • coord 𝕜 x h‖ = 1

Tactic:
simp only [norm_smul, RCLike.norm_coe_norm, coord_norm, mul_inv_cancel₀ (mt norm_eq_zero.mp h)]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := ⋯
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r * f z₁
⊢ 0 ≤ c * ‖z‖

pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := ⋯
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r * f z₁
⊢ ‖f z₁‖ * ‖z‖ ≤ c * ‖z‖

Tactic:
· exact mul_nonneg ((norm_nonneg _).trans norm_f_z₁) (norm_nonneg z)

Goal state after:
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := ⋯
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r * f z₁
⊢ ‖f z₁‖ * ‖z‖ ≤ c * ‖z‖
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
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ (setToL1 hT) ↑(indicatorConst 1 hs hμs x) = (T s) x

Tactic:
exact setToL1_simpleFunc_indicatorConst hT hs hμs.lt_top x

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

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝¹ : Countable ι
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : ∀ (i : ι), Integrable f (μ i)
h : Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i
⊢ HasFiniteIntegral f (Measure.sum μ)

Tactic:
rw [HasFiniteIntegral, lintegral_sum_measure]

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝¹ : Countable ι
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : ∀ (i : ι), Integrable f (μ i)
h : Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i
⊢ ∑' (i : ι), ∫⁻ (a : X), ‖f a‖ₑ ∂μ i < ∞
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ 0 C
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ 0 hT f = 0

Tactic:
· exact setToFun_undef hT hf

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

Goal state before:
pp: case neg
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_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = 0

Tactic:
· exact setToFun_undef hT hf

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
⊢ Bornology.IsBounded (s * t)

Tactic:
have key : IsCompact (closedBall (0 : ℝ≥0) Af ×ˢ closedBall (0 : ℝ≥0) Ag) :=
      IsCompact.prod (isCompact_closedBall _ _) (isCompact_closedBall _ _)

Goal state after:
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)
⊢ Bornology.IsBounded (s * t)
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
  by_cases hμ : IsFiniteMeasure μ
  · simp only [integral_eq_setToFun]
    exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _
  by_cases hc : c = 0
  · simp [hc, integral_zero]
  · simp [measureReal_def, (integrable_const_iff_isFiniteMeasure hc).not.2 hμ,
      integral_undef, MeasureTheory.not_isFiniteMeasure_iff.mp hμ]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f

Tactic:
by_cases hG'' : CompleteSpace G''

Goal state after:
case pos
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : CompleteSpace G''
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f

case neg
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : ¬CompleteSpace G''
⊢ setToFun μ T hT f ≤ setToFun μ T' hT' f
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : PartialOrder G''
inst✝² : IsOrderedAddMonoid G''
inst✝¹ : NormedSpace ℝ G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α → E
hG'' : CompleteSpace G''
hf : Integrable f μ
⊢ (L1.setToL1 hT) (Integrable.toL1 f hf) ≤ (L1.setToL1 hT') (Integrable.toL1 f hf)

Tactic:
exact L1.setToL1_mono_left' hT hT' hTT' _

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
⊢ 0 ≤ setToFun μ T hT f

Tactic:
by_cases hG'' : CompleteSpace G''

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
hG'' : CompleteSpace G''
⊢ 0 ≤ setToFun μ T hT f

case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
hG'' : ¬CompleteSpace G''
⊢ 0 ≤ setToFun μ T hT f
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
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ setToFun μ T hT ↑↑f

Tactic:
by
  by_cases hF : CompleteSpace F; swap
  · simp [setToFun, hF, continuous_const]
  simp_rw [L1.setToFun_eq_setToL1 hT]; exact ContinuousLinearMap.continuous _

Goal state after: