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 ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
]

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
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)
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 : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
h_contra : ∀ j ∈ s, ∀ k ∈ s, z j = z k
j : ι
hjs : j ∈ s
⊢ ∑ i ∈ s, w i * z j = ∑ i ∈ s, w i * z i

Tactic:
]

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 : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
h_contra : ∀ j ∈ s, ∀ k ∈ s, z j = z k
j : ι
hjs : j ∈ s
⊢ ∑ i ∈ s, w i * z j = ∑ i ∈ s, w i * z i
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ 1 ≤ p

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ 1 ≤ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm f p μ + eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm' f p.toReal μ + eLpNorm g p μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm' f p.toReal μ + eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm' f p.toReal μ + eLpNorm' g p.toReal μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
p : ℝ≥0∞
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
hp1 : 1 ≤ p
hp0 : ¬p = 0
hp_top : ¬p = ∞
hp1_real : 1 ≤ p.toReal
⊢ eLpNorm' (f + g) p.toReal μ ≤ eLpNorm' f p.toReal μ + eLpNorm' g p.toReal μ
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:
]

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
⊢ (∫⁻ (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.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
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 ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) =
    (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / q + 1 / r - 1 / q)

Tactic:
]

Goal state after:
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 ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r) =
    (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / q + 1 / r - 1 / q)
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
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ 1 * μ Set.univ ^ (1 / q)

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ 1 * μ Set.univ ^ (1 / q)
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 : MemLp f p μ
hg : MemLp g p μ
⊢ MemLp (f + -g) p μ

Tactic:
]

Goal state after:
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 : MemLp f p μ
hg : MemLp g p μ
⊢ MemLp (f + -g) 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
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ (q * q⁻¹) * μ Set.univ ^ (1 / q)

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) ≤ eLpNormEssSup f μ ^ (q * q⁻¹) * μ Set.univ ^ (1 / q)
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
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ q⁻¹ ≤ (eLpNormEssSup f μ ^ q * μ Set.univ) ^ q⁻¹

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ ∫⁻ (x : α), eLpNormEssSup f μ ^ q ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ q⁻¹ ≤ (eLpNormEssSup f μ ^ q * μ Set.univ) ^ q⁻¹
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₁
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ

Tactic:
]

Goal state after:
case h₁
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ
hq_pos : 0 < q
h_le : ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ ≤ eLpNormEssSup f μ ^ q * μ Set.univ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
⊢ ∏ x ∈ s, (z x ^ w x)⁻¹ ≤ ∏ i ∈ s, (z i)⁻¹ ^ w i

Tactic:
]

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
⊢ ∏ x ∈ s, (z x ^ w x)⁻¹ ≤ ∏ i ∈ s, (z i)⁻¹ ^ w i
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inr.inl.inr
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 : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hp0 : p ≠ 0
hpq : p ≤ ∞
hq0_lt : 0 < ∞
hp_top : p ≠ ∞
⊢ eLpNorm' f p.toReal μ ≤ eLpNorm f ∞ μ * μ Set.univ ^ p.toReal⁻¹

Tactic:
]

Goal state after:
case inr.inl.inr
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 : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hp0 : p ≠ 0
hpq : p ≤ ∞
hq0_lt : 0 < ∞
hp_top : p ≠ ∞
⊢ eLpNorm' f p.toReal μ ≤ eLpNorm f ∞ μ * μ Set.univ ^ p.toReal⁻¹
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inr.inr
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
hp0 : p ≠ 0
hq0_lt : 0 < q
hq_top : q ≠ ∞
hp_lt_top : p < ∞
hp_pos : 0 < p.toReal
⊢ eLpNorm' f p.toReal μ ≤ eLpNorm' f q.toReal μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)

Tactic:
]

Goal state after:
case inr.inr
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
hp0 : p ≠ 0
hq0_lt : 0 < q
hq_top : q ≠ ∞
hp_lt_top : p < ∞
hp_pos : 0 < p.toReal
⊢ eLpNorm' f p.toReal μ ≤ eLpNorm' f q.toReal μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case h1
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
x : ι
a✝ : x ∈ s
⊢ (z x ^ w x)⁻¹ ≤ (z x ^ w x)⁻¹

pp: case h1.hx
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
x : ι
a✝ : x ∈ s
⊢ 0 ≤ z x

Tactic:
]

Goal state after:
case h1.hx
ι : Type u
s : Finset ι
w z : ι → ℝ
hs : s.Nonempty
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 < z i
p_pos : 0 < ∏ i ∈ s, (z i)⁻¹ ^ w i
s_pos : 0 < ∑ i ∈ s, w i * (z i)⁻¹
this : (∑ i ∈ s, w i * (z i)⁻¹)⁻¹ ≤ (∏ i ∈ s, (z i)⁻¹ ^ w i)⁻¹
p_pos₂ : 0 < (∏ i ∈ s, z i ^ w i)⁻¹
x : ι
a✝ : x ∈ s
⊢ 0 ≤ z x
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
μ : Measure α
inst✝ : IsProbabilityMeasure μ
hf : AEStronglyMeasurable f μ
h_le_μ : eLpNorm' f p μ ≤ eLpNorm' f q μ
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
μ : Measure α
inst✝ : IsProbabilityMeasure μ
hf : AEStronglyMeasurable f μ
h_le_μ : eLpNorm' f p μ ≤ eLpNorm' f q μ
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ
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
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ 2 ^ (p - 1) * ∫⁻ (a : α), f a ^ p ∂μ < ∞ ∧ 2 ^ (p - 1) * ∫⁻ (a : α), g a ^ p ∂μ < ∞

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:
]

Goal state after:
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
⊢ 2 ^ (p - 1) * ∫⁻ (a : α), f a ^ p ∂μ < ∞ ∧ 2 ^ (p - 1) * ∫⁻ (a : α), g a ^ p ∂μ < ∞

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
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inl
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
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_nonpos : p ≤ 0
⊢ eLpNorm' f 0 μ < ∞

Tactic:
]

Goal state after:
case inl
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
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_nonpos : p ≤ 0
⊢ eLpNorm' f 0 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✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
hq_pos : 0 < q
⊢ eLpNorm' f q μ < ∞ ∧ μ Set.univ ^ (1 / p - 1 / q) < ∞ ∨ eLpNorm' f q μ = 0 ∨ μ Set.univ ^ (1 / p - 1 / q) = 0

Tactic:
]

Goal state after:
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
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
hq_pos : 0 < q
⊢ eLpNorm' f q μ < ∞ ∧ μ Set.univ ^ (1 / p - 1 / q) < ∞ ∨ eLpNorm' f q μ = 0 ∨ μ Set.univ ^ (1 / p - 1 / q) = 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case refine_3
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
⊢ (∑ i ∈ s, n⁻¹ * (w i / z i))⁻¹ ≤ (∏ i ∈ s, z i ^ w i) ^ n⁻¹

Tactic:
]

Goal state after:
case refine_3
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
⊢ (∑ i ∈ s, n⁻¹ * (w i / z i))⁻¹ ≤ (∏ i ∈ s, z i ^ w i) ^ n⁻¹
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
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
hq_pos : 0 < q
⊢ p ≤ q

Tactic:
]

Goal state after:
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
inst✝ : IsFiniteMeasure μ
hf : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm' f q μ < ∞
hp_nonneg : 0 ≤ p
hpq : p ≤ q
hp_pos : 0 < p
hq_pos : 0 < q
⊢ p ≤ q
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p = 0
⊢ AEStronglyMeasurable f μ

Tactic:
]

Goal state after:
case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p = 0
⊢ AEStronglyMeasurable f μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
⊢ ∏ i ∈ s, (z i ^ w i) ^ n⁻¹ = ∏ i ∈ s, z i ^ (w i / n)

Tactic:
]

Goal state after:
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
⊢ ∏ i ∈ s, (z i ^ w i) ^ n⁻¹ = ∏ i ∈ s, z i ^ (w i / n)
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
⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)

Tactic:
have hp0 : 0 ≤ p := le_of_lt hp0_lt

Goal state after:
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
⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)
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✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
⊢ MemLp f p μ

Tactic:
]

Goal state after:
case neg
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
⊢ MemLp f p μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
i : ι
hi : i ∈ s
⊢ z i ^ (w i * n⁻¹) = z i ^ (w i * n⁻¹)

Tactic:
]

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
⊢ 1 / r = 1 / q + 1 / r - 1 / q

Tactic:
]

Goal state after:
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
⊢ 1 / r = 1 / q + 1 / r - 1 / q
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∞
inst✝ : IsFiniteMeasure μ
hpq : q = ∞
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
⊢ q = ∞

Tactic:
]

Goal state after:
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∞
inst✝ : IsFiniteMeasure μ
hpq : q = ∞
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
⊢ q = ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
hq_top : q = ∞
⊢ eLpNorm f ∞ μ < ∞

Tactic:
]

Goal state after:
case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
hq_top : q = ∞
⊢ eLpNorm f ∞ μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f ∞ μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
hq_top : q = ∞
⊢ eLpNorm f ∞ μ < ∞

Tactic:
]

Goal state after:
case pos
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∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f ∞ μ < ∞
hp0 : p ≠ 0
hp_top : p = ∞
hq_top : q = ∞
⊢ eLpNorm f ∞ μ < ∞