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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
q : ℝ
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≤ μ
hq : 0 ≤ q
⊢ eLpNorm' f q ν ≤ eLpNorm' f q μ

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
q : ℝ
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≤ μ
hq : 0 ≤ q
⊢ eLpNorm' f q ν ≤ eLpNorm' f q μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≪ μ
⊢ eLpNormEssSup f ν ≤ eLpNormEssSup f μ

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≪ μ
⊢ eLpNormEssSup f ν ≤ eLpNormEssSup f μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≤ μ
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm f p ν ≤ eLpNorm f p μ

Tactic:
[

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ ν : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hμν : ν ≤ μ
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm f p ν ≤ eLpNorm f p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm f p (μ.restrict s) = eLpNorm f p μ

Tactic:
[

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ENormedAddMonoid ε
s : Set α
f : α → ε
hsf : Function.support f ⊆ s
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm f p (μ.restrict s) = eLpNorm f p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝⁵ : ENorm ε
R : Type u_7
inst✝⁴ : Semiring R
inst✝³ : IsDomain R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
c : R
hc : c ≠ 0
f : α → ε
⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ

Tactic:
[

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝⁵ : ENorm ε
R : Type u_7
inst✝⁴ : Semiring R
inst✝³ : IsDomain R
inst✝² : Module R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : Module.IsTorsionFree R ℝ≥0∞
c : R
hc : c ≠ 0
f : α → ε
⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
c : ℝ≥0∞
hc : c ≠ 0
f : α → ε
⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ

Tactic:
[

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : ENorm ε
c : ℝ≥0∞
hc : c ≠ 0
f : α → ε
⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case e_a.e_a
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ 1 / p.toReal = (1 / p).toReal

Tactic:
[

Goal state after:
case e_a.e_a
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ 1 / p.toReal = (1 / p).toReal
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ p.toNNReal.HolderTriple q.toNNReal r.toNNReal ↔ p.HolderTriple q r

Tactic:
[

Goal state after:
p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ p.toNNReal.HolderTriple q.toNNReal r.toNNReal ↔ p.HolderTriple q r
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
ε : Type u_8
inst✝ : ENorm ε
f : α → ε
μ ν : Measure α
⊢ eLpNorm f 1 (μ + ν) = eLpNorm f 1 μ + eLpNorm f 1 ν

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
ε : Type u_8
inst✝ : ENorm ε
f : α → ε
μ ν : Measure α
⊢ eLpNorm f 1 (μ + ν) = eLpNorm f 1 μ + eLpNorm f 1 ν
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ p.toNNReal.HolderConjugate q.toNNReal ↔ p.HolderConjugate q

Tactic:
[

Goal state after:
p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ p.toNNReal.HolderConjugate q.toNNReal ↔ p.HolderConjugate q
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
h : MemLp f p μ
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ) p μ < ∞

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
h : MemLp f p μ
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ) p μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
h : eLpNormEssSup f μ < ∞
⊢ ∀ᶠ (x : ℝ≥0) in map (fun x ↦ ‖f x‖₊) (ae μ), x ≤ (eLpNormEssSup f μ).toNNReal

Tactic:
[

Goal state after:
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
h : eLpNormEssSup f μ < ∞
⊢ ∀ᶠ (x : ℝ≥0) in map (fun x ↦ ‖f x‖₊) (ae μ), x ≤ (eLpNormEssSup f μ).toNNReal
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
h : p.HolderConjugate q
⊢ p * q = p + q

Tactic:
obtain rfl | hp := eq_or_ne p ∞
  · simp [ne_zero q ∞]
  obtain rfl | hq := eq_or_ne q ∞
  · simp [ne_zero p ∞]
  simpa [add_comm p, mul_add, add_mul, hp, hq, ne_zero p q, ne_zero q p, ENNReal.mul_inv_cancel,
    ENNReal.inv_mul_cancel_right] using congr(p * $((inv_add_inv_eq_one p q).symm) * q)

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

Goal state before:
pp: case inr.inr
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup F
f : α → F
inst✝¹ : Finite α
inst✝ : IsFiniteMeasure μ
hp₀ : p ≠ 0
hp : p ≠ ∞
⊢ ∃ c, ∀ (x : α), ‖f x‖ₑ ^ p.toReal ≤ ↑c

Tactic:
[

Goal state after:
case inr.inr
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup F
f : α → F
inst✝¹ : Finite α
inst✝ : IsFiniteMeasure μ
hp₀ : p ≠ 0
hp : p ≠ ∞
⊢ ∃ c, ∀ (x : α), ‖f x‖ₑ ^ p.toReal ≤ ↑c
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

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

Tactic:
[

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hg : AEStronglyMeasurable g (Measure.map f μ)
hf : AEMeasurable f μ
hp_zero : ¬p = 0
hp_top : p = ∞
⊢ eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

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

Tactic:
[

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

Goal state before:
pp: x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hx' : 0 < -x
hy' : 0 < -y
hxy' : -x ≠ -y
⊢ a • log x + b • log y = a • log (-x) + b • log (-y)

Tactic:
[

Goal state after:
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hx' : 0 < -x
hy' : 0 < -y
hxy' : -x ≠ -y
⊢ a • log x + b • log y = a • log (-x) + b • log (-y)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

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

Tactic:
[

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

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

Tactic:
[

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
β : Type u_8
mβ : MeasurableSpace β
f : α → β
g : β → ε
hf : MeasurableEmbedding f
⊢ MemLp g p (Measure.map f μ) ↔ MemLp (g ∘ f) p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
hp0 : ¬p.toReal = 0
hp : p ≠ 0
hp' : p ≠ ∞
⊢ liminf (fun i ↦ ∫⁻ (a : α), ↑‖f i a‖₊ ^ p.toReal ∂μ) atTop ≤ ↑R ^ p.toReal

Tactic:
[

Goal state after:
case neg
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
hp0 : ¬p.toReal = 0
hp : p ≠ 0
hp' : p ≠ ∞
⊢ liminf (fun i ↦ ∫⁻ (a : α), ↑‖f i a‖₊ ^ p.toReal ∂μ) atTop ≤ ↑R ^ p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp0 : ¬p.toReal = 0
hp : p ≠ 0
hp' : p ≠ ∞
hbdd : ∀ (n : ℕ), (∫⁻ (x : α), ‖f n x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹ ≤ ↑R
⊢ liminf (fun i ↦ ∫⁻ (a : α), ↑‖f i a‖₊ ^ p.toReal ∂μ) atTop ≤ ↑R ^ p.toReal

Tactic:
[

Goal state after:
case neg
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp0 : ¬p.toReal = 0
hp : p ≠ 0
hp' : p ≠ ∞
hbdd : ∀ (n : ℕ), (∫⁻ (x : α), ‖f n x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹ ≤ ↑R
⊢ liminf (fun i ↦ ∫⁻ (a : α), ↑‖f i a‖₊ ^ p.toReal ∂μ) atTop ≤ ↑R ^ p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp : ∞ ≠ 0
hbdd : ∀ (n : ℕ), eLpNorm (f n) ∞ μ ≤ ↑R
⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞

Tactic:
[

Goal state after:
case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp : ∞ ≠ 0
hbdd : ∀ (n : ℕ), eLpNorm (f n) ∞ μ ≤ ↑R
⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
hp : p ≠ 0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
hp' : ¬p = ∞
x : α
hx : liminf (fun n ↦ ‖f n x‖ₑ ^ p.toReal) atTop < ∞
hppos : 0 < p.toReal
this : liminf (fun n ↦ ‖f n x‖ₑ ^ p.toReal) atTop = liminf (fun n ↦ ‖f n x‖ₑ) atTop ^ p.toReal
⊢ liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞

Tactic:
this

Goal state after:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
hp : p ≠ 0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R
hp' : ¬p = ∞
x : α
hx : liminf (fun n ↦ ‖f n x‖ₑ) atTop ^ p.toReal < ∞
hppos : 0 < p.toReal
this : liminf (fun n ↦ ‖f n x‖ₑ ^ p.toReal) atTop = liminf (fun n ↦ ‖f n x‖ₑ) atTop ^ p.toReal
⊢ liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)

Tactic:
[

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i)

Tactic:
simp_rw [smul_nonneg_iff_pos_imp_nonneg, sub_pos, sub_nonneg, forall_and]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : IsStrictOrderedRing α
inst✝⁴ : AddCommGroup β
inst✝³ : LinearOrder β
inst✝² : IsOrderedAddMonoid β
inst✝¹ : Module α β
inst✝ : IsStrictOrderedModule α β
f : ι → α
g : ι → β
s : Set ι
⊢ MonovaryOn f g s ↔
    (∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → f x < f x_2 → g x ≤ g x_2) ∧
      ∀ (x : ι), x ∈ s → ∀ (x_2 : ι), x_2 ∈ s → g x < g x_2 → f x ≤ f x_2
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4
ι : Type u_1
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 ≤ z i
⊢ (∑ i ∈ s, w i * z i) / ∑ i ∈ s, w i = ∑ i ∈ s, (w i / ∑ i ∈ s, w i) * z i

Tactic:
[

Goal state after:
case e'_4
ι : Type u_1
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 ≤ z i
⊢ (∑ i ∈ s, w i * z i) / ∑ i ∈ s, w i = ∑ i ∈ s, (w i / ∑ i ∈ s, w i) * z i
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case convert_2
ι : Type u_1
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 ≤ z i
⊢ ∑ i ∈ s, w i / ∑ i ∈ s, w i = 1

Tactic:
[

Goal state after:
case convert_2
ι : Type u_1
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 ≤ z i
⊢ ∑ i ∈ s, w i / ∑ i ∈ s, w i = 1
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ ∫⁻ (c : α), funMulInvSnorm f p μ c ^ p ∂μ = 1

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ ∫⁻ (c : α), funMulInvSnorm f p μ c ^ p ∂μ = 1
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁷ : CommRing 𝕜
inst✝¹⁶ : LinearOrder 𝕜
inst✝¹⁵ : IsStrictOrderedRing 𝕜
inst✝¹⁴ : CommRing E
inst✝¹³ : LinearOrder E
inst✝¹² : IsStrictOrderedRing E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : LinearOrder F
inst✝⁹ : IsOrderedAddMonoid F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜 G
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module 𝕜 F
inst✝⁴ : Module E F
inst✝³ : IsScalarTower 𝕜 E F
inst✝² : SMulCommClass 𝕜 E F
inst✝¹ : IsOrderedModule 𝕜 F
inst✝ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : MonovaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (a * (a + b)) • f x • g x + (b * (a + b)) • f y • g y = a • f x • g x + b • f y • g y

Tactic:
[

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁷ : CommRing 𝕜
inst✝¹⁶ : LinearOrder 𝕜
inst✝¹⁵ : IsStrictOrderedRing 𝕜
inst✝¹⁴ : CommRing E
inst✝¹³ : LinearOrder E
inst✝¹² : IsStrictOrderedRing E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : LinearOrder F
inst✝⁹ : IsOrderedAddMonoid F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜 G
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module 𝕜 F
inst✝⁴ : Module E F
inst✝³ : IsScalarTower 𝕜 E F
inst✝² : SMulCommClass 𝕜 E F
inst✝¹ : IsOrderedModule 𝕜 F
inst✝ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : MonovaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (a * (a + b)) • f x • g x + (b * (a + b)) • f y • g y = a • f x • g x + b • f y • g y
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁸ : CommRing 𝕜
inst✝¹⁷ : LinearOrder 𝕜
inst✝¹⁶ : IsStrictOrderedRing 𝕜
inst✝¹⁵ : CommRing E
inst✝¹⁴ : LinearOrder E
inst✝¹³ : IsStrictOrderedRing E
inst✝¹² : AddCommGroup F
inst✝¹¹ : LinearOrder F
inst✝¹⁰ : IsOrderedAddMonoid F
inst✝⁹ : AddCommGroup G
inst✝⁸ : Module 𝕜 G
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : IsOrderedModule 𝕜 F
inst✝¹ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
inst✝ : IsOrderedModule 𝕜 E
hf : ConcaveOn 𝕜 s f
hg : ConcaveOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : AntivaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x • g x + b • f y • g y = (a * (a + b)) • f x • g x + (b * (a + b)) • f y • g y

Tactic:
[

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁸ : CommRing 𝕜
inst✝¹⁷ : LinearOrder 𝕜
inst✝¹⁶ : IsStrictOrderedRing 𝕜
inst✝¹⁵ : CommRing E
inst✝¹⁴ : LinearOrder E
inst✝¹³ : IsStrictOrderedRing E
inst✝¹² : AddCommGroup F
inst✝¹¹ : LinearOrder F
inst✝¹⁰ : IsOrderedAddMonoid F
inst✝⁹ : AddCommGroup G
inst✝⁸ : Module 𝕜 G
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : IsOrderedModule 𝕜 F
inst✝¹ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
inst✝ : IsOrderedModule 𝕜 E
hf : ConcaveOn 𝕜 s f
hg : ConcaveOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : AntivaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f x • g x + b • f y • g y = (a * (a + b)) • f x • g x + (b * (a + b)) • f y • g y
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hp1 : 1 ≤ p
⊢ (a ^ p + b ^ p) ^ (1 / p) ≤ a + b

Tactic:
lift a to NNReal using ha
  lift b to NNReal using hb
  exact_mod_cast NNReal.rpow_add_rpow_le_add a b hp1

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case refine_2
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ ((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ (∑ i ∈ s, w i * z i ^ p).toNNReal

Tactic:
[

Goal state after:
case refine_2
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ ((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ (∑ i ∈ s, w i * z i ^ p).toNNReal
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
G : Type u_4
inst✝¹¹ : CommRing 𝕜
inst✝¹⁰ : LinearOrder 𝕜
inst✝⁹ : IsStrictOrderedRing 𝕜
inst✝⁸ : CommRing E
inst✝⁷ : LinearOrder E
inst✝⁶ : IsStrictOrderedRing E
inst✝⁵ : AddCommGroup G
inst✝⁴ : Module 𝕜 G
inst✝³ : Module 𝕜 E
s : Set G
inst✝² : IsOrderedModule 𝕜 E
inst✝¹ : IsScalarTower 𝕜 E E
inst✝ : SMulCommClass 𝕜 E E
f : G → E
hf : ConvexOn 𝕜 s f
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
n : ℕ
⊢ ConvexOn 𝕜 s (f * f ^ n)

Tactic:
exact hf.mul (hf.pow hf₀ _) hf₀ (fun x hx ↦ pow_nonneg (hf₀ hx) _) <|
      (monovaryOn_self f s).pow_right₀ hf₀ n

Goal state after:
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
n : ℕ
⊢ ConvexOn 𝕜 univ fun x ↦ x ^ (n + n)

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
n : ℕ
⊢ ConvexOn 𝕜 univ fun x ↦ x ^ (n + n)