Theorem: MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum
File: Mathlib/MeasureTheory/Function/LpSpace.lean

tactic: have hp_pos : 0 < p := zero_lt_one.trans_le hp1
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i

tactic: suffices h_pow : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤ (∑' i, B i) ^ p
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i

tactic: have h_tsum_1 :
  ∀ g : ℕ → ℝ≥0∞, (∑' i, g i) = atTop.liminf fun n => ∑ i in Finset.range (n + 1), g i := by
  intro g
  rw [ENNReal.tsum_eq_liminf_sum_nat, ← liminf_nat_add _ 1]
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p

tactic: simp_rw [h_tsum_1 _]
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p

tactic: rw [← h_tsum_1]
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
⊢ (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) ≤
    liminf (fun n => ∑ i in Finset.range (n + 1), B i) atTop ^ p

tactic: have h_liminf_pow :
  (∫⁻ a, (atTop.liminf
    fun n => ∑ i in Finset.range (n + 1), (‖f (i + 1) a - f i a‖₊ : ℝ≥0∞)) ^ p ∂μ) =
    ∫⁻ a, atTop.liminf
      fun n => (∑ i in Finset.range (n + 1), (‖f (i + 1) a - f i a‖₊ : ℝ≥0∞)) ^ p ∂μ := by
  refine' lintegral_congr fun x => _
  have h_rpow_mono := ENNReal.strictMono_rpow_of_pos (zero_lt_one.trans_le hp1)
  have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2
  refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _
  all_goals isBoundedDefault
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
⊢ (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) ≤
    (∑' (i : ℕ), B i) ^ p

tactic: rw [h_liminf_pow]
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
h_liminf_pow :
  (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) =
    ∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ
⊢ (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) ≤
    (∑' (i : ℕ), B i) ^ p

tactic: refine' (lintegral_liminf_le' _).trans _
state_before: case h_pow
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
h_liminf_pow :
  (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) =
    ∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ
⊢ (∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ) ≤
    (∑' (i : ℕ), B i) ^ p

tactic: rwa [← ENNReal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div]
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_pow : (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i

tactic: simp [hp_pos]
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_pow : (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
⊢ 0 < 1 / p

tactic: intro g
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
⊢ ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop

tactic: rw [ENNReal.tsum_eq_liminf_sum_nat, ← liminf_nat_add _ 1]
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
g : ℕ → ℝ≥0∞
⊢ (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop

tactic: refine' lintegral_congr fun x => _
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
⊢ (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) =
    ∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ

tactic: have h_rpow_mono := ENNReal.strictMono_rpow_of_pos (zero_lt_one.trans_le hp1)
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
⊢ liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) atTop ^ p =
    liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) ^ p) atTop

tactic: have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
⊢ liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) atTop ^ p =
    liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) ^ p) atTop

tactic: refine' (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply _ _ _ _
state_before: α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) atTop ^ p =
    liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊) ^ p) atTop

tactic: all_goals isBoundedDefault
state_before: case refine'_1
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ IsBoundedUnder (fun x x_1 => x ≥ x_1) atTop fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊

case refine'_2
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) x - f i x‖₊

case refine'_3
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ IsBoundedUnder (fun x x_1 => x ≥ x_1) atTop fun x_1 =>
    ↑(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj)
      (∑ i in Finset.range (x_1 + 1), ↑‖f (i + 1) x - f i x‖₊)

case refine'_4
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop fun x_1 =>
    ↑(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj)
      (∑ i in Finset.range (x_1 + 1), ↑‖f (i + 1) x - f i x‖₊)

tactic: isBoundedDefault
state_before: case refine'_4
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
x : α
h_rpow_mono : StrictMono fun x => x ^ p
h_rpow_surj : Function.Surjective fun y => y ^ p
⊢ IsCoboundedUnder (fun x x_1 => x ≥ x_1) atTop fun x_1 =>
    ↑(StrictMono.orderIsoOfSurjective (fun x => x ^ p) h_rpow_mono h_rpow_surj)
      (∑ i in Finset.range (x_1 + 1), ↑‖f (i + 1) x - f i x‖₊)

tactic: exact fun n =>
  (Finset.aemeasurable_sum (Finset.range (n + 1)) fun i _ =>
        ((hf (i + 1)).sub (hf i)).ennnorm).pow_const
    _
state_before: case h_pow.refine'_1
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
h_liminf_pow :
  (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) =
    ∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ
⊢ ∀ (n : ℕ), AEMeasurable fun a => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p

tactic: exact liminf_le_of_frequently_le' (frequently_of_forall h)
state_before: case h_pow.refine'_2
α : Type u_1
E : Type u_2
F : Type ?u.8169395
G : Type ?u.8169398
m m0 : MeasurableSpace α
p✝ : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), (∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
h_tsum_1 : ∀ (g : ℕ → ℝ≥0∞), (∑' (i : ℕ), g i) = liminf (fun n => ∑ i in Finset.range (n + 1), g i) atTop
h_liminf_pow :
  (∫⁻ (a : α), liminf (fun n => ∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) atTop ^ p ∂μ) =
    ∫⁻ (a : α), liminf (fun n => (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p) atTop ∂μ
⊢ liminf (fun n => ∫⁻ (a : α), (∑ i in Finset.range (n + 1), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) atTop ≤
    (∑' (i : ℕ), B i) ^ p