mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:

Tactic:
‖f₁ a‖ ≤ ‖f₀ a‖ + ‖f₀ a - f₁ a‖ := norm_le_insert _ _
      _ ≤ ‖f₀ a‖ + g a := by gcongr

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

Goal state before:
pp: case neg
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : ¬a < b
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
rw [Ioo_eq_empty hab, Ico_eq_empty hab]

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1

Tactic:
by
  cases isEmpty_or_nonempty ι
  · simp [norm_mkPiAlgebra_of_empty]
  · refine le_antisymm norm_mkPiAlgebra_le ?_
    convert! ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1
    simp

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖1‖

Tactic:
refine le_antisymm ?_ ?_
  · refine opNorm_le_bound (norm_nonneg (1 : A)) ?_
    simp
  · convert! ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun _ => (1 : A)
    simp

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ↑s → Set X
u_open : ∀ (x : ↑s), IsOpen[inst✝³] (u x)
xu : ∀ (x : ↑s), ↑x ∈ u x
hu : ∀ (x : ↑s), IntegrableOn f (u x ∩ s) μ
T : Set ↑s
T_count : T.Countable
hT : s ⊆ ⋃ i ∈ T, u i
⊢ s ⊆ ⋃ u_1 ∈ u '' T, u_1

Tactic:
by rwa [biUnion_image]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
h_ind : ∀ (n : ℕ), U ⊆ ↑M + c ^ n • U
h_small : Tendsto (fun n ↦ c ^ n • U) atTop (𝓝 0).smallSets
⊢ FiniteDimensional 𝕜 Eᵤ

Tactic:
have hU_sub_M : U ⊆ M := by
    intro x hx
    choose m hm u hu h_eq using fun n ↦ h_ind n hx
    have hu_tendsto : Tendsto u atTop (𝓝 0) := by
      intro W hW
      exact (tendsto_smallSets_iff.mp h_small W hW).mono fun n hn ↦ hn (hu n)
    have hm_tendsto : Tendsto m atTop (𝓝 x) := by
      simpa [show m = fun n ↦ x - u n by grind] using! tendsto_const_nhds.sub hu_tendsto
    exact M.closed_of_finiteDimensional.mem_of_tendsto hm_tendsto (Eventually.of_forall hm)

Goal state after:
𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
h_ind : ∀ (n : ℕ), U ⊆ ↑M + c ^ n • U
h_small : Tendsto (fun n ↦ c ^ n • U) atTop (𝓝 0).smallSets
hU_sub_M : U ⊆ ↑M
⊢ FiniteDimensional 𝕜 Eᵤ
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ∫⁻ (x : α), ‖↑↑(f - g) x‖ₑ ∂μ = ∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ

Tactic:
← edist_zero_right

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ ∫⁻ (x : α), edist (↑↑(f - g) x) 0 ∂μ = ∫⁻ (x : α), edist (↑↑f x - ↑↑g x) 0 ∂μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case neg
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen[inst✝³] sᶜ
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ Integrable f (μ.restrict (s ∩ sᶜ))

pp: case neg
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen[inst✝³] sᶜ
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ MeasurableSet sᶜ

Tactic:
inter_compl_self,

Goal state after:
case neg
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen[inst✝³] sᶜ
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ Integrable f (μ.restrict ∅)

case neg
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsOpen[inst✝³] sᶜ
hf : LocallyIntegrableOn f s μ
x : X
h : x ∉ s
⊢ MeasurableSet sᶜ
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ∞
⊢ ‖T s‖ ≤ C * μ'.real s

Tactic:
have hμs : μ s < ∞ := (h s).trans_lt hμ's

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
μ' : Measure α
h : μ ≤ μ'
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
h' : ∀ (s : Set α), μ s = ∞ → μ' s = ∞
s : Set α
hs : MeasurableSet s
hμ's : μ' s < ∞
hμs : μ s < ∞
⊢ ‖T s‖ ≤ C * μ'.real s
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
s : Finset ι
f : ι → α → G
hf : ∀ i ∈ s, Integrable (f i) μ
⊢ ∫ (a : α), ∑ i ∈ s, f i a ∂μ = ∑ i ∈ s, ∫ (a : α), f i a ∂μ

Tactic:
simp only [integral_eq_setToFun]
  exact setToFun_finsetSum (dominatedFinMeasAdditive_weightedSMul _) s hf

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

Goal state before:
pp: case refine_2
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)
s : Set α
⊢ Tendsto (fun i ↦ ∫⁻ (x : α) in s, ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)

Tactic:
· simp_rw [← eLpNorm_one_eq_lintegral_enorm] at hF ⊢
    exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hF (fun _ ↦ zero_le)
      (fun _ ↦ eLpNorm_mono_measure _ Measure.restrict_le_self)

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ (fun a ↦ ENNReal.ofReal (-f a)) =ᵐ[μ] 0

Tactic:
refine hf.mono ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ∀ (x : α), 0 x ≤ f x → (fun a ↦ ENNReal.ofReal (-f a)) x = 0 x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
P : Type u_7
inst✝ : NormedAddCommGroup P
f : α → P
hf : Integrable f μ
⊢ ENNReal.ofReal (∫ (x : α), ‖f x‖ ∂μ) = ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
rw [integral_norm_eq_lintegral_enorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal]
  exact lt_top_iff_ne_top.mp (hasFiniteIntegral_iff_enorm.mpr hf.2)

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

Goal state before:
pp: 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 : CompleteSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
s : Set E
inst✝ : SeparableSpace ↑s
hs : range f ∪ {0} ⊆ s
⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s

Tactic:
exact Eventually.of_forall fun x => subset_closure (hs (Set.mem_union_left _ (mem_range_self _)))

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

Goal state before:
pp: case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
hf : MemLp (⇑f) p μ → motive ⇑f
hg : MemLp (⇑g) p μ → motive ⇑g
int_fg : MemLp (⇑(f + g)) p μ
⊢ motive ⇑(f + g)

Tactic:
rw [SimpleFunc.coe_add,
        memLp_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg

Goal state after:
case add
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
f g : α →ₛ E
hfg : Disjoint (support ⇑f) (support ⇑g)
hf : MemLp (⇑f) p μ → motive ⇑f
hg : MemLp (⇑g) p μ → motive ⇑g
int_fg : MemLp (⇑f) p μ ∧ MemLp (⇑g) p μ
⊢ motive ⇑(f + g)
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v : V
x : P
⊢ dist x (v +ᵥ x) = ‖v‖

Tactic:
rw [dist_comm, dist_vadd_left]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'

Tactic:
simp only [edist_nndist]

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ ↑(nndist (v +ᵥ p) (v' +ᵥ p')) ≤ ↑(nndist v v') + ↑(nndist p p')
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄

Tactic:
simp only [edist_nndist]

Goal state after:
V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ ↑(nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄)) ≤ ↑(nndist p₁ p₃) + ↑(nndist p₂ p₄)
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖c‖ / R * ‖d • x‖ = ‖d‖ * (‖c‖ / R * ‖x‖)

Tactic:
simp only [norm_smul]

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖c‖ / R * (‖d‖ * ‖x‖) = ‖d‖ * (‖c‖ / R * ‖x‖)
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
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i) μ
⊢ (setToFun μ T hT fun a ↦ ∑ i ∈ s, f i a) = ∑ i ∈ s, setToFun μ T hT (f i)

Tactic:
by
  convert! setToFun_finsetSum' hT s hf with a; simp

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

Goal state before:
pp: case pos
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
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT (-f) = -setToFun μ T hT f

Tactic:
rw [setToFun_eq hT hf, setToFun_eq hT hf.neg, Integrable.toL1_neg,
      (L1.setToL1 hT).map_neg]

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
⊢ ∫ (x : α), c ∂μ = μ.real univ • c

pp: case neg
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
⊢ ∫ (x : α), c ∂μ = μ.real univ • c

Tactic:
· simp only [integral_eq_setToFun]
    exact setToFun_const (dominatedFinMeasAdditive_weightedSMul _) _

Goal state after:
case neg
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
⊢ ∫ (x : α), c ∂μ = μ.real univ • c
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
hE : CompleteSpace E
mα : MeasurableSpace α
f : α → E
hf : StronglyMeasurable f
a : α
s : Set α
hs : MeasurableSet s
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0

Tactic:
· exact integral_zero_measure _

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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
h✝ : a ∉ s
⊢ ∫ (x : α), f x ∂0 = 0

Tactic:
· exact integral_zero_measure _

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
ι : Type u_1
inst✝ : Finite ι
v : Basis ι 𝕜 E
⊢ ∃ C > 0, ∀ {u : E →L[𝕜] F} (M : ℝ≥0), (∀ (i : ι), ‖u (v i)‖₊ ≤ M) → ‖u‖₊ ≤ C * M

Tactic:
cases nonempty_fintype ι
  exact
    ⟨max (Fintype.card ι • ‖v.equivFunL.toContinuousLinearMap‖₊) 1,
      zero_lt_one.trans_le (le_max_right _ _), fun {u} M hu =>
      (v.opNNNorm_le M hu).trans <| mul_le_mul_of_nonneg_right (le_max_left _ _) zero_le⟩

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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
s : Set E
inst✝ : SeparableSpace ↑s
hs : range f ∪ {0} ⊆ s
⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s

Tactic:
exact Eventually.of_forall fun x => subset_closure (hs (Set.mem_union_left _ (mem_range_self _)))

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
ι : Type u_1
inst✝ : Finite ι
v : Basis ι 𝕜 E
⊢ ∃ C > 0, ∀ {u : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ (i : ι), ‖u (v i)‖ ≤ M) → ‖u‖ ≤ C * M

Tactic:
by
  obtain ⟨C, hC, h⟩ := v.exists_opNNNorm_le (F := F)
  refine ⟨C, hC, ?_⟩
  intro u M hM H
  simpa using! h ⟨M, hM⟩ H

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case pos
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

Tactic:
simp [x_mem]

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_pos : ∀ (n : ℕ), 0 < δseq n
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
key :
  ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
x : α
x_mem : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x ≠ ∞

Tactic:
simp [x_mem]

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 1 ≤ (↑δ.toNNReal)⁻¹ * edist x y + (1 - infEDist y E * (↑δ.toNNReal)⁻¹) + infEDist x E * (↑δ.toNNReal)⁻¹

Tactic:
tsub_eq_zero_of_le (not_le.mp h_le).le,

Goal state after:
case neg
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
h : infEDist x E ≤ infEDist y E
h_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1
⊢ 1 ≤ (↑δ.toNNReal)⁻¹ * edist x y + 0 + infEDist x E * (↑δ.toNNReal)⁻¹
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
s t : Finset δ
hst : Disjoint s t
⊢ ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ

Tactic:
by
  rw [Finset.union_comm, lmarginal_union μ f hf hst.symm]

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

Goal state before:
pp: case pos
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
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
⊢ Tendsto (fun n ↦ setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))

Tactic:
have fs_int : ∀ n, Integrable (fs n) μ := fun n =>
    bound_integrable.mono' (fs_measurable n) (h_bound _)

Goal state after:
case pos
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
fs : ℕ → α → E
f : α → E
bound : α → ℝ
fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ
bound_integrable : Integrable bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) atTop (𝓝 (f a))
hF : CompleteSpace F
f_measurable : AEStronglyMeasurable f μ
fs_int : ∀ (n : ℕ), Integrable (fs n) μ
⊢ Tendsto (fun n ↦ setToFun μ T hT (fs n)) atTop (𝓝 (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✝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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
⊢ ∀ (n : ι), Integrable (F n) μ

Tactic:
refine fun n => bound_integrable.mono' (hF_meas n) ?_

Goal state after:
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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
n : ι
⊢ ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ ∑' (n : ι), bound n a