mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3.a
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
this :
  ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
i : ι
hi : i ∈ s
⊢ w i * f i = ↑(w i).toNNReal * ↑(f i).toNNReal

Tactic:
obtain hw | hw := eq_or_ne (w i) 0

Goal state after:
case e'_3.a.inl
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
this :
  ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
i : ι
hi : i ∈ s
hw : w i = 0
⊢ w i * f i = ↑(w i).toNNReal * ↑(f i).toNNReal

case e'_3.a.inr
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
this :
  ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
i : ι
hi : i ∈ s
hw : w i ≠ 0
⊢ w i * f i = ↑(w i).toNNReal * ↑(f i).toNNReal
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁵ : RingHomIsometric σ
inst✝⁴ : Fact (1 ≤ p)
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
⊢ compLpL p μ (c • L) = c • compLpL p μ L

Tactic:
ext1 f; exact smul_compLp c L f

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
h1 :
  ∀ (ns : ℕ → ℕ),
    StrictMono ns → ∃ ns', StrictMono ns' ∧ ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f (ns (ns' i)) ω) atTop (𝓝 (g ω))
⊢ TendstoInMeasure μ f atTop g

Tactic:
rw [tendstoInMeasure_iff_tendsto_toNNReal]

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
h1 :
  ∀ (ns : ℕ → ℕ),
    StrictMono ns → ∃ ns', StrictMono ns' ∧ ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f (ns (ns' i)) ω) atTop (𝓝 (g ω))
⊢ ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) (g x)}).toNNReal) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
h1 :
  ∀ (ns : ℕ → ℕ),
    StrictMono ns → ∃ ns', StrictMono ns' ∧ ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f (ns (ns' i)) ω) atTop (𝓝 (g ω))
ε : ℝ≥0∞
hε : 0 < ε
h2 : ¬Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) (g x)}).toNNReal) atTop (𝓝 0)
⊢ False

Tactic:
obtain ⟨δ, ns, hδ, hns, h3⟩ : ∃ (δ : ℝ≥0) (ns : ℕ → ℕ), 0 < δ ∧ StrictMono ns ∧
      ∀ n, δ ≤ (μ {x | ε ≤ edist (f (ns n) x) (g x)}).toNNReal := by
    obtain ⟨s, hs, h4⟩ := not_tendsto_iff_exists_frequently_notMem.1 h2
    obtain ⟨δ, hδ, h5⟩ := NNReal.nhds_zero_basis.mem_iff.1 hs
    obtain ⟨ns, hns, h6⟩ := extraction_of_frequently_atTop h4
    exact ⟨δ, ns, hδ, hns, fun n ↦ Set.notMem_Iio.1 (Set.notMem_subset h5 (h6 n))⟩

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
inst✝ : IsFiniteMeasure μ
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
g : α → E
h1 :
  ∀ (ns : ℕ → ℕ),
    StrictMono ns → ∃ ns', StrictMono ns' ∧ ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f (ns (ns' i)) ω) atTop (𝓝 (g ω))
ε : ℝ≥0∞
hε : 0 < ε
h2 : ¬Tendsto (fun i ↦ (μ {x | ε ≤ edist (f i x) (g x)}).toNNReal) atTop (𝓝 0)
δ : ℝ≥0
ns : ℕ → ℕ
hδ : 0 < δ
hns : StrictMono ns
h3 : ∀ (n : ℕ), δ ≤ (μ {x | ε ≤ edist (f (ns n) x) (g x)}).toNNReal
⊢ False
Module: Mathlib.Data.Fintype.CardEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
emb : Fintype (α ↪ β)
⊢ ‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖

Tactic:
rw [Subsingleton.elim emb Embedding.fintype]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
emb : Fintype (α ↪ β)
⊢ ‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p q : M →L[R] M
hp : IsIdempotentElem p
hq : IsIdempotentElem q
⊢ p = q ↔ (↑p).range = (↑q).range ∧ (↑p).ker = (↑q).ker

Tactic:
by
  simpa using LinearMap.IsIdempotentElem.ext_iff hp.toLinearMap hq.toLinearMap

Goal state after:
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hc : IsClosedMap f
h_cont : Continuous[inst✝¹, inst✝] f
⊢ IsStrictMap f

Tactic:
by
  rw [isStrictMap_iff_isQuotientMap_rangeFactorization]
  exact (hc.subtype_mk fun x => ⟨x, rfl⟩).isQuotientMap
    h_cont.rangeFactorization Set.rangeFactorization_surjective

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

Goal state before:
pp: case inl
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ t ≠ ∞
hs : μ s = ∞
⊢ μ.real s ≤ μ.real (s ∪ t)

pp: case inr
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ t ≠ ∞
hs : μ s < ∞
⊢ μ.real s ≤ μ.real (s ∪ t)

Tactic:
· simp [Measure.real, hs]

Goal state after:
case inr
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
h : μ t ≠ ∞
hs : μ s < ∞
⊢ μ.real s ≤ μ.real (s ∪ t)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : μ u ≠ ∞
⊢ (s ∩ t).Nonempty

Tactic:
inter_comm

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t u : Set α
hs : MeasurableSet s
h's : s ⊆ u
h't : t ⊆ u
h : μ.real u < μ.real t + μ.real s
hu : μ u ≠ ∞
⊢ (t ∩ s).Nonempty
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ AEStronglyMeasurable (fun x ↦ c) μ ∧ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ

Tactic:
and_iff_right this,

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ MemLp (fun x ↦ ‖f x‖ₑ ^ p.toReal) 1 μ

Tactic:
exact hf.enorm_rpow hp_ne_zero hp_ne_top

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case empty
α : Type u_1
ε : Type u_5
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
ι : Type u_8
m : MeasurableSpace α
f : α → ε
μ : ι → Measure α
⊢ Integrable f (∑ i ∈ ∅, μ i) ↔ ∀ i ∈ ∅, Integrable f (μ i)

Tactic:
simp [*]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case insert
α : Type u_1
ε : Type u_5
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
ι : Type u_8
m : MeasurableSpace α
f : α → ε
μ : ι → Measure α
a✝² : ι
s✝ : Finset ι
a✝¹ : a✝² ∉ s✝
a✝ : Integrable f (∑ i ∈ s✝, μ i) ↔ ∀ i ∈ s✝, Integrable f (μ i)
⊢ Integrable f (∑ i ∈ insert a✝² s✝, μ i) ↔ ∀ i ∈ insert a✝² s✝, Integrable f (μ i)

Tactic:
simp [*]

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
μ : Measure α
l : Filter α
𝕜 : Type u_7
inst✝² : NormedAddCommGroup 𝕜
inst✝¹ : SMulZeroClass 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : α → E
hf : IntegrableAtFilter f l μ
c : 𝕜
⊢ IntegrableAtFilter (c • f) l μ

Tactic:
by
  rcases hf with ⟨s, sl, hs⟩
  exact ⟨s, sl, hs.smul c⟩

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
hconv : StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x
x : X
⊢ (l.limUnder fun x_1 ↦ f x_1 x) = Function.extend Subtype.val (fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x) (fun x ↦ e) x

Tactic:
by_cases hx : x ∈ conv

Goal state after:
case pos
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
hconv : StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x
x : X
hx : x ∈ conv
⊢ (l.limUnder fun x_1 ↦ f x_1 x) = Function.extend Subtype.val (fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x) (fun x ↦ e) x

case neg
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace E
inst✝² : Countable ι
l : Filter ι
inst✝¹ : l.IsCountablyGenerated
f : ι → X → E
hE : Nonempty E
inst✝ : IsCompletelyMetrizableSpace E
hf : ∀ (i : ι), StronglyMeasurable (f i)
hl : l.NeBot
e : E := Classical.choice hE
conv : Set X := {x | ∃ c, Tendsto (fun x_1 ↦ f x_1 x) l (𝓝 c)}
mconv : MeasurableSet conv
hconv : StronglyMeasurable fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x
x : X
hx : x ∉ conv
⊢ (l.limUnder fun x_1 ↦ f x_1 x) = Function.extend Subtype.val (fun x ↦ l.limUnder fun x_1 ↦ f x_1 ↑x) (fun x ↦ e) x
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
x : Eᵤ
hx : x ∈ U
m : ℕ → Eᵤ
hm : ∀ (n : ℕ), m n ∈ ↑M
u : ℕ → Eᵤ
hu : ∀ (n : ℕ), u n ∈ c ^ n • U
h_eq : ∀ (n : ℕ), (fun x1 x2 ↦ x1 + x2) (m n) (u n) = x
⊢ Tendsto u atTop (𝓝 0)

Tactic:
intro W

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
x : Eᵤ
hx : x ∈ U
m : ℕ → Eᵤ
hm : ∀ (n : ℕ), m n ∈ ↑M
u : ℕ → Eᵤ
hu : ∀ (n : ℕ), u n ∈ c ^ n • U
h_eq : ∀ (n : ℕ), (fun x1 x2 ↦ x1 + x2) (m n) (u n) = x
W : Set Eᵤ
⊢ W ∈ 𝓝 0 → W ∈ map u atTop
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 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
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'
m : (i : ι) → E i
⊢ ‖{ toFun := fun m ↦ { toFun := fun x ↦ (f x) m, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖f‖ * ∏ i, ‖m i‖) ⋯,
          map_update_add' := ⋯, map_update_smul' := ⋯ }
        m‖ ≤
    ‖f‖ * ∏ i, ‖m i‖

Tactic:
dsimp only [MultilinearMap.coe_mk]

Goal state after:
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
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'
m : (i : ι) → E i
⊢ ‖{ toFun := fun x ↦ (f x) m, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (‖f‖ * ∏ i, ‖m i‖) ⋯‖ ≤ ‖f‖ * ∏ i, ‖m i‖
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
p : ℝ≥0∞
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : μ s ≠ ∞
h's : MeasurableSet s
⊢ (μ.restrict s) univ ≠ ∞

Tactic:
by simpa using hs

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
p : ℝ≥0∞
f : X → E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
⊢ MemLp f p (μ.restrict s)

Tactic:
obtain rfl | h := s.eq_empty_or_nonempty
  · simp
  · exact hmono.memLp_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h)
      hs.measure_lt_top.ne hs.measurableSet

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 E
inst✝ : SecondCountableTopologyEither X 𝕜
f : X → 𝕜
hf : ContinuousOn f K
g : X → E
hg : IntegrableOn g A μ
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
⊢ IntegrableOn (fun x ↦ f x • g x) A μ

Tactic:
rcases IsCompact.exists_bound_of_continuousOn hK hf with ⟨C, hC⟩
  exact hg.bdd_smul C ((hf.mono hAK).aestronglyMeasurable hA)
    (ae_restrict_of_forall_mem hA fun x hx => hC x (hAK hx))

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
⊢ ‖(T + T') s‖ ≤ max C C' * (μ + ν).real s

Tactic:
have hνs : ν s < ∞ := (Measure.le_add_left le_rfl s).trans_lt hsf

Goal state after:
α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ ‖(T + T') s‖ ≤ max C C' * (μ + ν).real s
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive (c • μ) T C
s : Set α
a✝ : MeasurableSet s
hcμs : ¬c = 0 ∧ μ s = ∞
⊢ μ s = ∞

Tactic:
exact hcμs.2

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ IsDenseEmbedding Subtype.val

Tactic:
apply simpleFunc.isUniformEmbedding.isDenseEmbedding

Goal state after:
α : Type u_1
E : Type u_4
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝ : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
this✝¹ : MeasurableSpace E := ⋯
this✝ : BorelSpace E
⊢ DenseRange Subtype.val
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)

Tactic:
have : SeparableSpace ((range (g : α → G) ∪ {0}) ∩ { y | 0 ≤ y } : Set G) := by
    apply IsSeparable.separableSpace
    apply IsSeparable.mono _ Set.inter_subset_left
    exact
      (Lp.stronglyMeasurable (g : Lp G p μ)).isSeparable_range.union
        (finite_singleton _).isSeparable

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
f : α →ₛ F
hf : 0 ≤ᵐ[μ] ⇑f
⊢ 0 ≤ integral μ f

Tactic:
integral_eq

Goal state after:
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
f : α →ₛ F
hf : 0 ≤ᵐ[μ] ⇑f
⊢ 0 ≤ ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case hg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] ℝ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
ae_eq₂ : ⇑(toSimpleFunc f).negPart =ᵐ[μ] ⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))
⊢ Integrable (⇑(SimpleFunc.map norm (toSimpleFunc (negPart f)))) μ

Tactic:
exact (SimpleFunc.integrable f).neg_part.congr ae_eq₂

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T T' : Set α → E →L[ℝ] G''
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α →ₛ E
hf : Integrable (⇑f) μ
i : E
a✝ : i ∈ f.range
h0 : i = 0
⊢ (T (⇑f ⁻¹' {i})) i ≤ (T' (⇑f ⁻¹' {i})) i

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
G'' : Type u_8
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T T' : Set α → E →L[ℝ] G''
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α →ₛ E
hf : Integrable (⇑f) μ
i : E
a✝ : i ∈ f.range
h0 : ¬i = 0
⊢ (T (⇑f ⁻¹' {i})) i ≤ (T' (⇑f ⁻¹' {i})) i

Tactic:
· simp [h0]

Goal state after:
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
G'' : Type u_8
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
T T' : Set α → E →L[ℝ] G''
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
f : α →ₛ E
hf : Integrable (⇑f) μ
i : E
a✝ : i ∈ f.range
h0 : ¬i = 0
⊢ (T (⇑f ⁻¹' {i})) i ≤ (T' (⇑f ⁻¹' {i})) i
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁶ : NormedAddCommGroup G''
inst✝⁵ : PartialOrder G''
inst✝⁴ : IsOrderedAddMonoid G''
inst✝³ : NormedSpace ℝ G''
inst✝² : NormedAddCommGroup G'
inst✝¹ : PartialOrder G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α →ₛ G'
hf : 0 ≤ f
hfi : Integrable (⇑f) μ
i : G'
hi : i ∈ f.range
h0 : i = 0
⊢ 0 ≤ (T (⇑f ⁻¹' {i})) i

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁶ : NormedAddCommGroup G''
inst✝⁵ : PartialOrder G''
inst✝⁴ : IsOrderedAddMonoid G''
inst✝³ : NormedSpace ℝ G''
inst✝² : NormedAddCommGroup G'
inst✝¹ : PartialOrder G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α →ₛ G'
hf : 0 ≤ f
hfi : Integrable (⇑f) μ
i : G'
hi : i ∈ f.range
h0 : ¬i = 0
⊢ 0 ≤ (T (⇑f ⁻¹' {i})) i

Tactic:
· simp [h0]

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁶ : NormedAddCommGroup G''
inst✝⁵ : PartialOrder G''
inst✝⁴ : IsOrderedAddMonoid G''
inst✝³ : NormedSpace ℝ G''
inst✝² : NormedAddCommGroup G'
inst✝¹ : PartialOrder G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α →ₛ G'
hf : 0 ≤ f
hfi : Integrable (⇑f) μ
i : G'
hi : i ∈ f.range
h0 : ¬i = 0
⊢ 0 ≤ (T (⇑f ⁻¹' {i})) i