mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (∫⁻ (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:
let r := p * q / (q - p)

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)
⊢ (∫⁻ (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.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case e'_4
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
p : ℝ≥0∞
hp : p ≠ 0
h'p : p < 1
this : p ∈ Set.Ioo 0 1
⊢ p.LpAddConst = 2 ^ (1 / p.toReal - 1)

Tactic:
simp only [LpAddConst, if_pos this]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : α → E
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp : 1 ≤ p
⊢ eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
simpa [LpAddConst_of_one_le hp] using eLpNorm_sub_le' hf hg p

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

Goal state before:
pp: case refine_2
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'
this : DecidableEq ι
i : ι
⊢ ∀ (s : Finset ι),
    i ∉ s →
      ((∀ i ∈ s, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ s, f i a) p μ) →
        (∀ i_1 ∈ insert i s, MemLp (f i_1) p μ) → MemLp (fun a ↦ ∑ i ∈ insert i s, f i a) p μ

Tactic:
s

Goal state after:
case refine_2
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'
this : DecidableEq ι
i : ι
s : Finset ι
⊢ i ∉ s →
    ((∀ i ∈ s, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ s, f i a) p μ) →
      (∀ i_1 ∈ insert i s, MemLp (f i_1) p μ) → MemLp (fun a ↦ ∑ i ∈ insert i s, f i a) p μ
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✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ≥0∞
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
⊢ eLpNorm f p μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)

Tactic:
by
  obtain rfl | hp0 := eq_or_ne p 0
  · simp
  have hq0_lt : 0 < q := hp0.pos.trans_le hpq
  obtain rfl | hq_top := eq_or_ne q ∞
  · simp only [_root_.div_zero, one_div, ENNReal.toReal_top, sub_zero]
    obtain rfl | hp_top := eq_or_ne p ∞
    · simp
    rw [eLpNorm_eq_eLpNorm' hp0 hp_top]
    have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top
    refine (eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ hp_pos).trans (le_of_eq ?_)
    congr
    exact one_div _
  have hp_lt_top : p < ∞ := hpq.trans_lt (lt_top_iff_ne_top.mpr hq_top)
  have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_lt_top.ne
  rw [eLpNorm_eq_eLpNorm' hp0 hp_lt_top.ne, eLpNorm_eq_eLpNorm' hq0_lt.ne.symm hq_top]
  have hpq_real : p.toReal ≤ q.toReal := ENNReal.toReal_mono hq_top hpq
  exact eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ hp_pos hpq_real hf

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

Goal state before:
pp: α : Type u_2
inst✝ : MeasurableSpace α
p q r : ℝ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp0_ne : p ≠ 0
hp0 : 0 ≤ p
hq0_lt : 0 < q
hq0_ne : q ≠ 0
h_one_div_r : 1 / r = 1 / p - 1 / q
p2 : ℝ := q / p
q2 : ℝ := p2.conjExponent
hp2q2 : p2.HolderConjugate q2
hpp2 : p * p2 = q
hpq2 : p * q2 = r
⊢ (∫⁻ (a : α), f a ^ (p * p2) ∂μ) ^ (1 / p2 * (1 / p)) * (∫⁻ (a : α), g a ^ (p * q2) ∂μ) ^ (1 / q2 * (1 / p)) =
    (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)

Tactic:
simp_rw [div_mul_div_comm, mul_one, mul_comm p2, mul_comm q2, hpp2, hpq2]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ q) (B ^ q)
⊢ ∃ C ≤ A * B, HasSum (fun i ↦ f i * g i) C

Tactic:
obtain ⟨H₁, H₂⟩ := summable_and_inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable

Goal state after:
ι : Type u
f g : ι → ℝ≥0
A B : ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : HasSum (fun i ↦ f i ^ p) (A ^ p)
hg : HasSum (fun i ↦ g i ^ q) (B ^ q)
H₁ : Summable fun i ↦ f i * g i
H₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)
⊢ ∃ C ≤ A * B, HasSum (fun i ↦ f i * g i) C
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
let q : ℝ := p / (p - 1)

Goal state after:
case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case h₁
ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ p
pos : 0 < p
s : Finset ι
⊢ Summable fun i ↦ f i ^ p

pp: case h₁
ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ p
pos : 0 < p
s : Finset ι
⊢ Summable fun i ↦ g i ^ p

Tactic:
exacts [hf, hg]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
h_eq : (fun n ↦ eLpNorm (↑↑(f n) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) = fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ
⊢ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) fi (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)

Tactic:
h_eq

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
h_eq : (fun n ↦ eLpNorm (↑↑(f n) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) p μ) = fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ
⊢ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0) ↔ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
⊢ Tendsto f fi (𝓝 (MemLp.toLp f_lim f_lim_ℒp)) ↔ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)

Tactic:
rw [tendsto_Lp_iff_tendsto_eLpNorm']
  suffices h_eq :
      (fun n => eLpNorm (⇑(f n) - ⇑(MemLp.toLp f_lim f_lim_ℒp)) p μ) =
        (fun n => eLpNorm (⇑(f n) - f_lim) p μ) by
    rw [h_eq]
  exact funext fun n => eLpNorm_congr_ae (EventuallyEq.rfl.sub (MemLp.coeFn_toLp f_lim_ℒp))

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case neg
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 :
  ↑(∑ i ∈ s, (w i).toNNReal * (f i).toNNReal) ≤
    ↑((∑ i ∈ s, (w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, (w i).toNNReal * (f i).toNNReal ^ p) ^ p⁻¹)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
rw [coe_mul] at this

Goal state after:
case neg
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 :
  ↑(∑ i ∈ s, (w i).toNNReal * (f i).toNNReal) ≤
    ↑((∑ i ∈ s, (w i).toNNReal) ^ (1 - p⁻¹)) * ↑((∑ i ∈ s, (w i).toNNReal * (f i).toNNReal ^ p) ^ p⁻¹)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
let q : ℝ := p / (p - 1)

Goal state after:
case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∈ {a | ¬Tendsto (fun i ↦ f (ns i) a) atTop (𝓝 (g a))} → x ∈ s

Tactic:
rw [Set.mem_ofPred_eq, ← @Classical.not_not (x ∈ s), not_imp_not]

Goal state after:
case h
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∉ s → Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' s)
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
i : ι
this : ∀ᶠ (x : 𝕜) in 𝓝 0, x • eval i '' s ⊆ t i
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : 𝕜⦄, x ∈ {y | ‖y‖ < r} → x • eval i '' s ⊆ t i
c : 𝕜
hc₀ : 0 < ‖c‖
hc : ‖c‖ < r
⊢ ∃ c, c ≠ 0 ∧ ∀ (c' : 𝕜), ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i

Tactic:
refine ⟨c, norm_pos_iff.1 hc₀, fun c' hle x hx ↦ ?_⟩

Goal state after:
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' s)
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
i : ι
this : ∀ᶠ (x : 𝕜) in 𝓝 0, x • eval i '' s ⊆ t i
r : ℝ
hr₀ : 0 < r
hr : ∀ ⦃x : 𝕜⦄, x ∈ {y | ‖y‖ < r} → x • eval i '' s ⊆ t i
c : 𝕜
hc₀ : 0 < ‖c‖
hc : ‖c‖ < r
c' : 𝕜
hle : ‖c'‖ ≤ ‖c‖
x : (i : ι) → E i
hx : x ∈ s
⊢ c' • x i ∈ t i
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
t : R
h : ‖t‖ < 1
⊢ (1 - t)⁻¹ʳ = ↑(Units.oneSub t h)⁻¹

Tactic:
← inverse_unit (Units.oneSub t h),

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
t : R
h : ‖t‖ < 1
⊢ (1 - t)⁻¹ʳ = (↑(Units.oneSub t h))⁻¹ʳ
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : RingHomIsometric σ₁₂
hn : ‖f‖ = 0
x : E
⊢ ‖f‖ * ‖x‖ = 0

Tactic:
by rw [hn, zero_mul]

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real (t ∪ s) + μ.real (t ∩ s) = μ.real s + μ.real t

Tactic:
measureReal_union_add_inter₀ hs h₂ h₁,

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real t + μ.real s = μ.real s + μ.real t
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSingletonClass α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset α
f : α → E
⊢ IntegrableOn f (↑s) μ

Tactic:
← (s : Set α).biUnion_of_singleton

Goal state after:
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSingletonClass α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset α
f : α → E
⊢ IntegrableOn f (⋃ x ∈ ↑s, {x}) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
G : Type wG
G' : Type wG'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
f : G →L[𝕜] G'
⊢ ‖(ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖

Tactic:
let : Unique ι := uniqueOfSubsingleton i

Goal state after:
𝕜 : Type u
ι : Type v
G : Type wG
G' : Type wG'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
f : G →L[𝕜] G'
this : Unique ι := uniqueOfSubsingleton i
⊢ ‖(ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : CommMonoid E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : ContinuousMul E
L : SummationFilter ι
inst✝³ : L.NeBot
inst✝² : L.filter.IsCountablyGenerated
inst✝¹ : IsCompletelyPseudoMetrizableSpace E
inst✝ : Countable ι
μ : Measure X
f g : ι → X → E
hg_meas : ∀ (i : ι), StronglyMeasurable (g i)
hg_eq_f : ∀ (i : ι), f i =ᵐ[μ] g i
x : X
⊢ (∀ (i : ι), f i x = g i x) → ∏'[L] (i : ι), f i x = ∏'[L] (i : ι), g i x

Tactic:
h_eq

Goal state after:
X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : CommMonoid E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : ContinuousMul E
L : SummationFilter ι
inst✝³ : L.NeBot
inst✝² : L.filter.IsCountablyGenerated
inst✝¹ : IsCompletelyPseudoMetrizableSpace E
inst✝ : Countable ι
μ : Measure X
f g : ι → X → E
hg_meas : ∀ (i : ι), StronglyMeasurable (g i)
hg_eq_f : ∀ (i : ι), f i =ᵐ[μ] g i
x : X
h_eq : ∀ (i : ι), f i x = g i x
⊢ ∏'[L] (i : ι), f i x = ∏'[L] (i : ι), g i x
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)
⊢ ‖f - g‖ = (∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ).toReal

Tactic:
rw [norm_def]

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)
⊢ (∫⁻ (a : α), ‖↑↑(f - g) a‖ₑ ∂μ).toReal = (∫⁻ (x : α), ‖↑↑f x - ↑↑g x‖ₑ ∂μ).toReal
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
inst✝ : IsLocallyFiniteMeasure μ
p : ℝ≥0∞
hf : MemLp f p μ
hp : 1 ≤ p
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : μ U < ∞
this : Fact (μ U < ∞)
⊢ MemLp f 1 (μ.restrict U)

Tactic:
apply (hf.restrict U).mono_exponent hp

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup E
μ : Measure X
f : X → E
⊢ LocallyIntegrable (-f) μ ↔ LocallyIntegrable f μ

Tactic:
by
  simp [← locallyIntegrableOn_univ]

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

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝¹ : NormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : X → E
c : 𝕜
⊢ LocallyIntegrable (c • f) μ ↔ c = 0 ∨ LocallyIntegrable f μ

Tactic:
by
  simp [← locallyIntegrableOn_univ]

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

Goal state before:
pp: case hf
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → E
hf : LocallyIntegrable f μ
g : X → 𝕜
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := ⋯
hK : IsCompact K
this : (K.indicator fun x ↦ g x • f x) = fun x ↦ g x • f x
⊢ Integrable (K.indicator f) μ

Tactic:
rw [integrable_indicator_iff hK.measurableSet]

Goal state after:
case hf
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → E
hf : LocallyIntegrable f μ
g : X → 𝕜
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ g x • f x) = fun x ↦ g x • f x
⊢ IntegrableOn f K μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case hφ
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := ⋯
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ Integrable (K.indicator f) μ

Tactic:
rw [integrable_indicator_iff hK.measurableSet]

Goal state after:
case hφ
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ IntegrableOn f K μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : Fintype ι
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ContinuousMultilinearMap 𝕜 E G
⊢ ‖f‖ = 0 ↔ f = 0

Tactic:
simp [← (opNorm_nonneg f).ge_iff_eq', opNorm_le_iff le_rfl, ContinuousMultilinearMap.ext_iff]

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace ε
inst✝⁴ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝³ : PseudoMetrizableSpace ε
inst✝² : LinearOrder X
inst✝¹ : OrderTop X
inst✝ : CompactIccSpace X
⊢ Integrable f μ ↔ IntegrableAtFilter f atBot μ ∧ LocallyIntegrable f μ

Tactic:
constructor
  · exact fun hf ↦ ⟨hf.integrableAtFilter _, hf.locallyIntegrable⟩
  · refine fun h ↦ integrable_iff_integrableAtFilter_cocompact.mpr ⟨?_, h.2⟩
    exact h.1.filter_mono cocompact_le_atBot

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
s : Set α
x✝ : MeasurableSet s
hμs : (c • μ) s = ∞
⊢ μ s = ∞

Tactic:
Measure.smul_apply,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
s : Set α
x✝ : MeasurableSet s
hμs : c • μ s = ∞
⊢ μ s = ∞
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_zero : c ≠ 0
hT : FinMeasAdditive μ T
s : Set α
x✝ : MeasurableSet s
hμs : μ s = ∞
⊢ (c • μ) s = ∞

Tactic:
Measure.smul_apply,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_zero : c ≠ 0
hT : FinMeasAdditive μ T
s : Set α
x✝ : MeasurableSet s
hμs : μ s = ∞
⊢ c • μ s = ∞
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
⊢ Measurable fun x ↦ ‖f x - y₀‖

Tactic:
simp only [← dist_eq_norm]

Goal state after:
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
⊢ Measurable fun x ↦ dist (f x) y₀