mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case pos
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
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

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
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
·

Goal state after:
case pos
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
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case pos
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 = ∞
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

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 = ∞
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
·

Goal state after:
case pos
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 = ∞
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case hg
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ AEMeasurable (fun a ↦ ∏ i ∈ s, f i a ^ q i) μ

pp: case hp
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ p i₀

pp: case hq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ 1 - p i₀

pp: case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1

Tactic:
·

Goal state after:
case hg
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ AEMeasurable (fun a ↦ ∏ i ∈ s, f i a ^ q i) μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case mpr
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
⊢ (∃ j ∈ s, ∃ k ∈ s, z j ≠ z k) → ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i

Tactic:
·

Goal state after:
case mpr
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
⊢ (∃ j ∈ s, ∃ k ∈ s, z j ≠ z k) → ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case hp
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ p i₀

pp: case hq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ 1 - p i₀

pp: case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1

Tactic:
·

Goal state after:
case hp
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ p i₀
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case hq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ 1 - p i₀

pp: case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1

Tactic:
·

Goal state after:
case hq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ 0 ≤ 1 - p i₀
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1

Tactic:
·

Goal state after:
case hpq
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ p i₀ + (1 - p i₀) = 1
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case inl
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
⊢ eLpNorm (f + g) 0 μ ≤ LpAddConst 0 * (eLpNorm f 0 μ + eLpNorm g 0 μ)

pp: case inr
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
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)

Tactic:
·

Goal state after:
case inl
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
⊢ eLpNorm (f + g) 0 μ ≤ LpAddConst 0 * (eLpNorm f 0 μ + eLpNorm g 0 μ)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case inr.inl
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
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)

pp: case inr.inr
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 : 1 ≤ p
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)

Tactic:
·

Goal state after:
case inr.inl
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
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)
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
⊢ p.LpAddConst = 2 ^ (1 / p.toReal - 1)

pp: case inr.inl.convert_3
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
⊢ p.toReal ≤ 1

Tactic:
·

Goal state after:
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
⊢ p.LpAddConst = 2 ^ (1 / p.toReal - 1)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case inr.inl.convert_3
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
⊢ p.toReal ≤ 1

Tactic:
·

Goal state after:
case inr.inl.convert_3
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
⊢ p.toReal ≤ 1
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: case inr.inr
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 : 1 ≤ p
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)

Tactic:
·

Goal state after:
case inr.inr
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 : 1 ≤ p
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case insert.inr
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
⊢ ∫⁻ (a : α), ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i ∈ insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
· have hpi₀ : 0 ≤ 1 - p i₀ := by
        simp_rw [sub_nonneg, ← hp, single_le_sum h2p (mem_insert_self ..)]
      have h2pi₀ : 1 - p i₀ ≠ 0 := by
        rwa [sub_ne_zero, ne_comm]
      let q := fun i ↦ p i / (1 - p i₀)
      have hq : ∑ i ∈ s, q i = 1 := by
        rw [← Finset.sum_div, ← sum_insert_sub hi₀, hp, div_self h2pi₀]
      have h2q : ∀ i ∈ s, 0 ≤ q i :=
        fun i hi ↦ div_nonneg (h2p i <| mem_insert_of_mem hi) hpi₀
      calc ∫⁻ a, ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ
          = ∫⁻ a, f i₀ a ^ p i₀ * ∏ i ∈ s, f i a ^ p i ∂μ := by simp [hi₀]
        _ = ∫⁻ a, f i₀ a ^ p i₀ * (∏ i ∈ s, f i a ^ q i) ^ (1 - p i₀) ∂μ := by
            simp [q, ← ENNReal.prod_rpow_of_nonneg hpi₀, ← ENNReal.rpow_mul,
              div_mul_cancel₀ (h := h2pi₀)]
        _ ≤ (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * (∫⁻ a, ∏ i ∈ s, f i a ^ q i ∂μ) ^ (1 - p i₀) := by
            apply ENNReal.lintegral_mul_norm_pow_le
            · exact hf i₀ <| mem_insert_self ..
            · exact s.aemeasurable_fun_prod fun i hi ↦ (hf i <| mem_insert_of_mem hi).pow_const _
            · exact h2p i₀ <| mem_insert_self ..
            · exact hpi₀
            · apply add_sub_cancel
        _ ≤ (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * (∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ q i) ^ (1 - p i₀) := by
            gcongr -- behavior of gcongr is heartbeat-dependent, which makes code really fragile...
            exact ih (fun i hi ↦ hf i <| mem_insert_of_mem hi) hq h2q
        _ = (∫⁻ a, f i₀ a ∂μ) ^ p i₀ * ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i := by
            simp [q, ← ENNReal.prod_rpow_of_nonneg hpi₀, ← ENNReal.rpow_mul,
              div_mul_cancel₀ (h := h2pi₀)]
        _ = ∏ i ∈ insert i₀ s, (∫⁻ a, f i a ∂μ) ^ p i := by simp [hi₀]

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

Goal state before:
pp: case refine_1
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 ∈ ∅, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ ∅, f i a) p μ

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
⊢ ∀ (a : ι) (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ s, f i a) p μ) →
        (∀ i ∈ insert a s, MemLp (f i) p μ) → MemLp (fun a_3 ↦ ∑ i ∈ insert a s, f i a_3) p μ

Tactic:
·

Goal state after:
case refine_1
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 ∈ ∅, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ ∅, f i a) p μ
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
⊢ ∀ (a : ι) (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ s, f i a) p μ) →
        (∀ i ∈ insert a s, MemLp (f i) p μ) → MemLp (fun a_3 ↦ ∑ i ∈ insert a s, f i a_3) p μ

Tactic:
·

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
⊢ ∀ (a : ι) (s : Finset ι),
    a ∉ s →
      ((∀ i ∈ s, MemLp (f i) p μ) → MemLp (fun a ↦ ∑ i ∈ s, f i a) p μ) →
        (∀ i ∈ insert a s, MemLp (f i) p μ) → MemLp (fun a_3 ↦ ∑ i ∈ insert a s, f i a_3) 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✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hpq : 0 ≤ q
⊢ eLpNorm f 0 μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / ENNReal.toReal 0 - 1 / q.toReal)

pp: case 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
⊢ eLpNorm f p μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / p.toReal - 1 / q.toReal)

Tactic:
·

Goal state after:
case inl
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≥0∞
hf : AEStronglyMeasurable f μ
hpq : 0 ≤ q
⊢ eLpNorm f 0 μ ≤ eLpNorm f q μ * μ Set.univ ^ (1 / ENNReal.toReal 0 - 1 / q.toReal)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case refine_1
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
⊢ ∀ i ∈ insertNone s, AEMeasurable (fun t ↦ i.elim (g t) fun j ↦ f j t) μ

pp: case refine_2
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
⊢ ∑ i ∈ insertNone s, i.elim q p = 1

pp: case refine_3
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
⊢ ∀ i ∈ insertNone s, 0 ≤ i.elim q p

Tactic:
·

Goal state after:
case refine_1
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
⊢ ∀ i ∈ insertNone s, AEMeasurable (fun t ↦ i.elim (g t) fun j ↦ f j t) μ
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case refine_1.none
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
hi : none ∈ insertNone s
⊢ AEMeasurable (fun t ↦ none.elim (g t) fun j ↦ f j t) μ

pp: case refine_1.some
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
i : ι
hi : some i ∈ insertNone s
⊢ AEMeasurable (fun t ↦ (some i).elim (g t) fun j ↦ f j t) μ

Tactic:
·

Goal state after:
case refine_1.none
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
hi : none ∈ insertNone s
⊢ AEMeasurable (fun t ↦ none.elim (g t) fun j ↦ f j t) μ
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case refine_1.some
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
i : ι
hi : some i ∈ insertNone s
⊢ AEMeasurable (fun t ↦ (some i).elim (g t) fun j ↦ f j t) μ

Tactic:
·

Goal state after:
case refine_1.some
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
i : ι
hi : some i ∈ insertNone s
⊢ AEMeasurable (fun t ↦ (some i).elim (g t) fun j ↦ f j t) μ