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

Goal state before:
pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
⊢ ∀ (s : (i : ι) → Set (α i)),
    (univ.pi s).Nonempty → (pi' μ).toOuterMeasure (univ.pi s) ≤ ∏ i, (μ i).toOuterMeasure (s i)

Tactic:
intro s _

Goal state after:
case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
s : (i : ι) → Set (α i)
a✝ : (univ.pi s).Nonempty
⊢ (pi' μ).toOuterMeasure (univ.pi s) ≤ ∏ i, (μ i).toOuterMeasure (s i)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s = (log (1 + s) - log 1) / s

Tactic:
log_one,

Goal state after:
case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s = (log (1 + s) - 0) / s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - log 1) / (p * s)

Tactic:
log_one,

Goal state after:
case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - 0) / (p * s)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - log 1) / (p * s)

Tactic:
log_one,

Goal state after:
case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - 0) / (p * s)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + s) / s = (log (1 + s) - log 1) / s

Tactic:
log_one,

Goal state after:
case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + s) / s = (log (1 + s) - 0) / s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - log 1) / (p * s)

Tactic:
log_one,

Goal state after:
case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - 0) / (p * s)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s = (log (1 + s) - log 1) / s

Tactic:
log_one,

Goal state after:
case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) / s = (log (1 + s) - 0) / s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + s) / s = (log (1 + s) - log 1) / s

Tactic:
log_one,

Goal state after:
case e'_3
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + s) / s = (log (1 + s) - 0) / s
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - log 1) / (p * s)

Tactic:
log_one,

Goal state after:
case e'_4
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : 0 < s
⊢ log (1 + p * s) / (p * s) = (log (1 + p * s) - 0) / (p * s)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hf : MeasurableEmbedding f
hp_zero : ¬p = 0
hp : p = ∞
⊢ eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ

pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hf : MeasurableEmbedding f
hp_zero : ¬p = 0
hp : ¬p = ∞
⊢ eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ

Tactic:
· simp_rw [hp, eLpNorm_exponent_top]
    exact hf.essSup_map_measure

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hf : MeasurableEmbedding f
hp_zero : ¬p = 0
hp : ¬p = ∞
⊢ eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ j ∈ s, ∀ k ∈ s, z j = z k
k : ι
hk : k ∈ s
⊢ z k ^ 1 = 1 * z k

Tactic:
rpow_one,

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ j ∈ s, ∀ k ∈ s, z j = z k
k : ι
hk : k ∈ s
⊢ z k = 1 * z k
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: z₁ z₂ p : ℝ≥0∞
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)

Tactic:
by_cases p_zero : p = 0

Goal state after:
case pos
z₁ z₂ p : ℝ≥0∞
p_zero : p = 0
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)

case neg
z₁ z₂ p : ℝ≥0∞
p_zero : ¬p = 0
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
r : ℝ := p * q / (q - p)
⊢ (∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
have hpqr : 1 / p = 1 / q + 1 / r := by simp [field]

Goal state after:
case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
r : ℝ := p * q / (q - p)
hpqr : 1 / p = 1 / q + 1 / r
⊢ (∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
s : Finset ι
g : α → ℝ≥0∞
f : ι → α → ℝ≥0∞
hg : AEMeasurable g μ
hf : ∀ i ∈ s, AEMeasurable (f i) μ
q : ℝ
p : ι → ℝ
hpq : q + ∑ i ∈ s, p i = 1
hq : 0 ≤ q
hp : ∀ i ∈ s, 0 ≤ p i
this :
  ∫⁻ (t : α), ∏ j ∈ insertNone s, (j.elim (g t) fun j ↦ f j t) ^ j.elim q p ∂μ ≤
    ∏ j ∈ insertNone s, (∫⁻ (t : α), j.elim (g t) fun j ↦ f j t ∂μ) ^ j.elim q p
⊢ ∫⁻ (a : α), g a ^ q * ∏ i ∈ s, f i a ^ p i ∂μ ≤ (∫⁻ (a : α), g a ∂μ) ^ q * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
by
    simpa using this

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: α : Type u_1
ε' : Type u_4
m : MeasurableSpace α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddCommMonoid ε'
p : ℝ≥0∞
μ : Measure α
inst✝ : ContinuousAdd ε'
ι : Type u_5
s : Finset ι
f : ι → α → ε'
hf : ∀ i ∈ s, MemLp (f i) p μ
⊢ MemLp (∑ i ∈ s, f i) p μ

Tactic:
convert! memLp_finsetSum s hf using 1
  ext x
  simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
inst✝ : IsProbabilityMeasure μ
⊢ eLpNormEssSup f μ * μ Set.univ ^ (1 / q) = eLpNormEssSup f μ

Tactic:
by simp [measure_univ]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ≥0∞
hpq : p ≤ q
inst✝ : IsProbabilityMeasure μ
hf : AEStronglyMeasurable f μ
⊢ eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal) = eLpNorm f q μ

Tactic:
by simp [measure_univ]

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

Goal state before:
pp: case hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ AEMeasurable (fun a ↦ 2 ^ (p - 1) * f a ^ p) μ

Tactic:
· fun_prop

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.inr.refine_1
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ (∑ i ∈ s, f i ^ p) / (∑ i ∈ s, f i ^ p) ^ 1 = 1

Tactic:
rpow_one,

Goal state after:
case inr.inr.refine_1
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ (∑ i ∈ s, f i ^ p) / ∑ i ∈ s, f i ^ p = 1
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.inr.refine_2
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ (∑ i ∈ s, g i ^ q) / (∑ i ∈ s, g i ^ q) ^ 1 = 1

Tactic:
rpow_one,

Goal state after:
case inr.inr.refine_2
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ (∑ i ∈ s, g i ^ q) / ∑ i ∈ s, g i ^ q = 1
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case inl
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hp0 : 0 ≤ 0
hp1 : 0 ≤ 1
⊢ 1 ≤ (2 ^ 1)⁻¹ * 2

Tactic:
rpow_one,

Goal state after:
case inl
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hp0 : 0 ≤ 0
hp1 : 0 ≤ 1
⊢ 1 ≤ 2⁻¹ * 2
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case pos
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
f : ι → α → E
l : Filter ι
g : α → E
h : ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε = ∞
h1 : Tendsto (fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}) l (𝓝 0)
⊢ (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) ≤ fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}

Tactic:
intro n

Goal state after:
case pos
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
f : ι → α → E
l : Filter ι
g : α → E
h : ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε = ∞
h1 : Tendsto (fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}) l (𝓝 0)
n : ι
⊢ (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) n ≤ (fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}) n
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
f : ι → α → E
l : Filter ι
g : α → E
h : ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)
⊢ TendstoInMeasure μ f l g

Tactic:
by
  intro ε hε
  by_cases hε_top : ε = ∞
  · have h1 : Tendsto (fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}) l (𝓝 0) := h 1 (by simp) (by simp)
    refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds h1 (fun _ ↦ zero_le) ?_
    intro n
    simp only [hε_top]
    gcongr
    simp
  · exact h ε hε hε_top

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_pow_liminf : liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p = liminf (fun n ↦ eLpNorm' (f n) p μ ^ p) atTop
⊢ liminf (fun i ↦ ∫⁻ (a : α), ‖f i a‖ₑ ^ p ∂μ) atTop = liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p

Tactic:
rw [h_pow_liminf]

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_pow_liminf : liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p = liminf (fun n ↦ eLpNorm' (f n) p μ ^ p) atTop
⊢ liminf (fun i ↦ ∫⁻ (a : α), ‖f i a‖ₑ ^ p ∂μ) atTop = liminf (fun n ↦ eLpNorm' (f n) p μ ^ p) atTop
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case refine_2
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ p) (B ^ p)
hp' : p ≠ 0
H₁ : Summable fun i ↦ (f i + g i) ^ p
H₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p)
⊢ HasSum (fun i ↦ (f i + g i) ^ p) (((∑' (i : ι), (f i + g i) ^ p) ^ (1 / p)) ^ p)

Tactic:
· simpa only [rpow_self_rpow_inv hp'] using H₁.hasSum

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0
⊢ (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0

Tactic:
simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
        sum_eq_zero_iff_of_nonneg] using H

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ 0) (p / 0) μ < ∞

Tactic:
by_cases p_zero : p = 0

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
p_zero : p = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ 0) (p / 0) μ < ∞

case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
p_zero : ¬p = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ 0) (p / 0) μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
⊢ eLpNorm (fun x ↦ 1) (p / 0) μ < ∞

Tactic:
by_cases p_zero : p = 0

Goal state after:
case pos
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
p_zero : p = 0
⊢ eLpNorm (fun x ↦ 1) (p / 0) μ < ∞

case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
p_zero : ¬p = 0
⊢ eLpNorm (fun x ↦ 1) (p / 0) μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
⊢ ∫⁻ (a : α), liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) atTop ^ p ∂μ =
    ∫⁻ (a : α), liminf (fun n ↦ (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p) atTop ∂μ

Tactic:
refine lintegral_congr fun x => ?_

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), ∑' (i : ℕ), g i = liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), g i) atTop
x : α
⊢ liminf (fun n ↦ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖ₑ) atTop ^ p =
    liminf (fun n ↦ (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖ₑ) ^ p) atTop
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
h : μ ≤ c • ν
inst✝ : Fact (1 ≤ p)
f : ↥(Lp E p ν)
⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ.toReal ≤ c.toReal ^ (1 / p).toReal * ‖f‖ₑ.toReal

Tactic:
ENNReal.toReal_rpow,

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
hc : c ≠ ∞
h : μ ≤ c • ν
inst✝ : Fact (1 ≤ p)
f : ↥(Lp E p ν)
⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ.toReal ≤ (c ^ (1 / p).toReal).toReal * ‖f‖ₑ.toReal
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: case pos.refine_1
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → essSup (fun x ↦ ‖(f n - f m_1) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (B N).toReal

Tactic:
intro n

Goal state after:
case pos.refine_1
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → essSup (fun x ↦ ‖(f n - f m_1) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
n : ℕ
⊢ ∀ (m N : ℕ), N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (B N).toReal
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝⁴ : RingHomInvPair σ₁₂ σ₂₁
inst✝³ : RingHomInvPair σ₂₁ σ₁₂
inst✝² : RingHomIsometric σ₂₁
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : Nontrivial E
e : E ≃SL[σ₁₂] F
⊢ 1 = ‖↑e.symm ∘SL ↑e‖

Tactic:
e.coe_symm_comp_coe,

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝⁴ : RingHomInvPair σ₁₂ σ₂₁
inst✝³ : RingHomInvPair σ₂₁ σ₁₂
inst✝² : RingHomIsometric σ₂₁
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : Nontrivial E
e : E ≃SL[σ₁₂] F
⊢ 1 = ‖ContinuousLinearMap.id 𝕜 E‖
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case refine_2
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
aux : (F →L[𝕜] G) →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F →L[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := fun g ↦ { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ },
    map_add' := ⋯, map_smul' := ⋯ }
U : Set (ContinuousMultilinearMap 𝕜 E F)
V : Set ((i : ι) → E i)
W : Set G
hU : IsVonNBounded 𝕜 (U, V, W).1
hV : IsVonNBounded 𝕜 (U, V, W).2.1
hW : (U, V, W).2.2 ∈ 𝓝 0
⊢ ∀ x ∈ {f | ∀ x ∈ (image2 (fun f v ↦ f v) U V, W).1, f x ∈ (image2 (fun f v ↦ f v) U V, W).2},
    aux x ∈ {f | ∀ x ∈ (U, V, W).1, f x ∈ {f | MapsTo ⇑f (U, V, W).2.1 (U, V, W).2.2}}

Tactic:
· exact fun g hg f hf m hm ↦ hg _ <| mem_image2_of_mem hf hm

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
c : ℝ≥0∞
⊢ (c • μ).real s = c.toReal * μ.real s

Tactic:
by
  simp [Measure.real]

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

Goal state before:
pp: case ha
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (s \ t) ≠ ∞

pp: case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (t \ s) ≠ ∞

Tactic:
· exact measure_ne_top_of_subset sdiff_subset h₁

Goal state after:
case hb
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ (t \ s) ≠ ∞