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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hro_lt : 0 < r
hrp : r < p
hpqr : 1 / r = 1 / p + 1 / q
⊢ eLpNorm' (fun x ↦ ↑c • ‖f x‖ * ‖g x‖) r μ ≤ ↑c * eLpNorm' f p μ * eLpNorm' g q μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hro_lt : 0 < r
hrp : r < p
hpqr : 1 / r = 1 / p + 1 / q
⊢ eLpNorm' (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm' f p μ * eLpNorm' g q μ

Tactic:
by

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
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
by

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 : ℝ
hpqr : p.HolderTriple q r
⊢ ∑ i ∈ s, (f i * g i) ^ r ≤ (∑ i ∈ s, f i ^ p) ^ (r / p) * (∑ i ∈ s, g i ^ q) ^ (r / q)

Tactic:
by

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 : ℝ
hpqr : p.HolderTriple q r
⊢ (∑ i ∈ s, (f i * g i) ^ r) ^ (1 / r) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
⊢ p⁻¹ + (1 - p⁻¹)⁻¹⁻¹ = 1

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : ∞.HolderTriple q r
hpqr : 1 / r = 1 / ∞ + 1 / q
⊢ r = q

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
⊢ p ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
hp₀ : p ≠ 0
⊢ 1 - p⁻¹ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
p : ℝ
hp : 1 ≤ p
w f : ι → ℝ≥0
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hp : 0 < p.toReal
hpqr✝ : p.HolderTriple ∞ r
hpqr : 1 / r = 1 / p + 1 / ∞
⊢ r = p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ ∀ (s : Finset ι), ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
this : (1 / r).toReal = (1 / p + 1 / q).toReal
⊢ 1 / p ≠ ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h :
  ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / q)
⊢ 1 / q = 1 - 1 / p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
H₁ : ∀ (s : Finset ι), ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)
⊢ BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i * g i) ^ r)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
this : (1 / r).toReal = (1 / p + 1 / q).toReal
⊢ 1 / q ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧
    ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
⊢ 1 / r.toReal = 1 / p.toReal + 1 / q.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ (Summable fun i ↦ f i * g i) ∧
    ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
⊢ 0 < 1 / r.toReal

Tactic:
by

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)
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)
⊢ A = (∑' (i : ι), f i ^ p) ^ (1 / p)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
hr : 0 < r.toReal
hr₁ : 0 < r
hr₂ : r < ∞
⊢ 0 < 1 / q.toReal

Tactic:
by

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)
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)
hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p)
⊢ B = (∑' (i : ι), g i ^ q) ^ (1 / q)

Tactic:
by

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:
by

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:
by
  obtain ⟨H₁, H₂⟩ := summable_and_inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable
  have hA : A = (∑' i : ι, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]
  have hB : B = (∑' i : ι, g i ^ q) ^ (1 / q) := by
    rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]
  refine ⟨∑' i, f i * g i, ?_, ?_⟩
  · simpa [hA, hB] using H₂
  · simpa only [rpow_self_rpow_inv hpq.ne_zero] using H₁.hasSum

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ≥0
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : p.HolderConjugate q
hp₁ : 1 / p * p = 1
⊢ 1 / q * p = p - 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
⊢ p + q - q ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
A : p + q - q ≠ 0
⊢ ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
⊢ IsGreatest ((fun g ↦ ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1}) ((∑ i ∈ s, f i ^ p) ^ (1 / p))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr : p.HolderTriple q r
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : ¬p = 1
⊢ 1 < p

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
p q r : ℝ≥0∞
f : α → E
g : α → F
b : E → F → G
c : ℝ≥0
hf : MemLp f p μ
hg : MemLp g q μ
h : AEStronglyMeasurable (fun x ↦ b (f x) (g x)) μ
hb : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr : p.HolderTriple q r
⊢ MemLp (fun x ↦ b (f x) (g x)) r μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : α → E
p : ℝ≥0∞
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
x✝ : α
⊢ ‖φ x✝ • f x✝‖₊ ≤ 1 * ‖φ x✝‖₊ * ‖f x✝‖₊

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : α → E
p : ℝ≥0∞
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
⊢ eLpNorm (φ • f) p μ ≤ eLpNorm φ ∞ μ * eLpNorm f p μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p : ℝ≥0∞
f : α → E
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
x✝ : α
⊢ ‖φ x✝ • f x✝‖₊ ≤ 1 * ‖φ x✝‖₊ * ‖f x✝‖₊

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p : ℝ≥0∞
f : α → E
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
⊢ eLpNorm (φ • f) p μ ≤ eLpNorm φ p μ * eLpNorm f ∞ μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
H₁ :
  ∀ (s : Finset ι), ∑ i ∈ s, (f i + g i) ^ p ≤ ((∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)) ^ p
⊢ BddAbove (Set.range fun s ↦ ∑ i ∈ s, (f i + g i) ^ p)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p q r : ℝ
f : α → E
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
x✝ : α
⊢ ‖φ x✝ • f x✝‖₊ ≤ 1 * ‖φ x✝‖₊ * ‖f x✝‖₊

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p q r : ℝ
f : α → E
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
⊢ eLpNorm' (φ • f) p μ ≤ eLpNorm' φ q μ * eLpNorm' f r μ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p q r : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
hpqr : p.HolderTriple q r
x✝ : α
⊢ ‖φ x✝ • f x✝‖₊ ≤ 1 * ‖φ x✝‖₊ * ‖f x✝‖₊

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p q r : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
hpqr : p.HolderTriple q r
⊢ eLpNorm (φ • f) r μ ≤ eLpNorm φ p μ * eLpNorm f q μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : ¬p = 1
hp1_lt : 1 < p
hpq : p.HolderConjugate p.conjExponent
h0 : ∫⁻ (a : α), (f + g) a ^ p ∂μ = 0
⊢ 0 < 1 / p

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s✝ : Finset ι
i : ι
s : Finset ι
hi : i ∉ s
ih : (∀ i ∈ s, MemLp (f i) (p i) μ) → MemLp (∏ i ∈ s, f i) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ
hf : ∀ i_1 ∈ cons i s hi, MemLp (f i_1) (p i_1) μ
⊢ (p i)⁻¹ + (∑ i ∈ s, (p i)⁻¹)⁻¹⁻¹ = (∑ i ∈ cons i s hi, (p i)⁻¹)⁻¹⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
⊢ (Summable fun i ↦ (f i + g i) ^ p) ∧
    (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ s, MemLp (f i) (p i) μ
⊢ MemLp (∏ i ∈ s, f i) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ s, MemLp (f i) (p i) μ
⊢ MemLp (fun ω ↦ ∏ i ∈ s, f i ω) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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)
⊢ A = (∑' (i : ι), f i ^ p) ^ (1 / p)

Tactic:
by

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✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
p : ℝ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_no_pow : ∫⁻ (a : α), ‖f_lim a‖ₑ ^ p ∂μ = ∫⁻ (a : α), liminf (fun m ↦ ‖f m a‖ₑ ^ p) atTop ∂μ
⊢ (∫⁻ (a : α), ‖f_lim a‖ₑ ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ) ^ (1 / p)

Tactic:
h_no_pow

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
p : ℝ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
h_no_pow : ∫⁻ (a : α), ‖f_lim a‖ₑ ^ p ∂μ = ∫⁻ (a : α), liminf (fun m ↦ ‖f m a‖ₑ ^ p) atTop ∂μ
⊢ (∫⁻ (a : α), liminf (fun m ↦ ‖f m a‖ₑ ^ p) atTop ∂μ) ^ (1 / p) =
    (∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ) ^ (1 / p)