Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case refine_1
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
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ‖f n x‖ₑ

pp: case refine_2
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
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ‖f n x‖ₑ

pp: case refine_3
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
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦ (ENNReal.orderIsoRpow p.toReal hppos) ‖f x_1 x‖ₑ

pp: case refine_4
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
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦ (ENNReal.orderIsoRpow p.toReal hppos) ‖f x_1 x‖ₑ

Tactic:
<;>

Goal state after:
case refine_1
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
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ‖f n x‖ₑ

case refine_2
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
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun n ↦ ‖f n x‖ₑ

case refine_3
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
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦ (ENNReal.orderIsoRpow p.toReal hppos) ‖f x_1 x‖ₑ

case refine_4
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
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) atTop fun x_1 ↦ (ENNReal.orderIsoRpow p.toReal hppos) ‖f x_1 x‖ₑ