mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ

Tactic:
by_cases hp0 : p = 0

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

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
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: 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
⊢ ∫⁻ (a : α), g a ^ q * ∏ i ∈ s, f i a ^ p i ∂μ ≤ (∫⁻ (a : α), g a ∂μ) ^ q * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
suffices
    ∫⁻ t, ∏ j ∈ insertNone s, Option.elim j (g t) (fun j ↦ f j t) ^ Option.elim j q p ∂μ
    ≤ ∏ j ∈ insertNone s, (∫⁻ t, Option.elim j (g t) (fun j ↦ f j t) ∂μ) ^ Option.elim j q p by
    simpa using this

Goal state after:
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
⊢ ∫⁻ (t : α), ∏ j ∈ insertNone s, (j.elim (g t) fun j ↦ f j t) ^ j.elim q p ∂μ ≤
    ∏ j ∈ insertNone s, (∫⁻ (t : α), j.elim (g t) fun j ↦ f j t ∂μ) ^ j.elim q p
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

Tactic:
rintro (_ | i) hi
    · exact hg
    · refine hf i ?_
      simpa using hi

Goal state after:
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:
simp_rw [inv_mul_eq_div n ((w _) / (z _)), div_right_comm _ _ n]

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)
⊢ (∑ x ∈ s, w x / n / z x)⁻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≥0∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
⊢ MemLp f p μ

Tactic:
by_cases hp0 : p = 0

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
⊢ MemLp f p μ

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: case inr.calc_1.e_f
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
i : ι
⊢ ↑(w i * f i) = ↑(w i ^ (1 - p⁻¹) * w i ^ p⁻¹ * f i)

Tactic:
← rpow_of_add_eq _ one_ne_zero,

Goal state after:
case inr.calc_1.e_f
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
i : ι
⊢ ↑(w i * f i) = ↑(w i ^ 1 * f i)

case inr.calc_1.e_f
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
i : ι
⊢ 1 - p⁻¹ + p⁻¹ = 1
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : α → E
p : ℝ≥0∞
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
⊢ eLpNorm (φ • f) p μ ≤ eLpNorm φ ∞ μ * eLpNorm f p μ

Tactic:
simpa using! (eLpNorm_le_eLpNorm_top_mul_eLpNorm p φ hf (· • ·) 1
    (.of_forall fun _ => by simpa using! nnnorm_smul_le _ _) :)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ (∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ) ^ (1 / p) ≤ liminf (fun n ↦ eLpNorm' (f n) p μ) atTop

Tactic:
rw [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : SeminormedAddGroup E
f : ℕ → α → E
p : ℝ
hp_pos : 0 < p
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ ∫⁻ (a : α), liminf (fun x ↦ ‖f x a‖ₑ ^ p) atTop ∂μ ≤ liminf (fun n ↦ eLpNorm' (f n) p μ) atTop ^ p
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : ↥(Lp E p μ)
⊢ Tendsto (fun b ↦ dist (f b) f_lim) fi (𝓝 0) ↔ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)

Tactic:
dist_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
f_lim : ↥(Lp E p μ)
⊢ Tendsto (fun b ↦ (eLpNorm (↑↑(f b) - ↑↑f_lim) p μ).toReal) fi (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (↑↑(f n) - ↑↑f_lim) p μ) fi (𝓝 0)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
ι : Type u_4
inst✝¹ : Nonempty ι
inst✝ : SemilatticeSup ι
hp : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
⊢ Tendsto (fun n ↦ dist (f n.1) (f n.2)) atTop (𝓝 0) ↔ Tendsto (fun n ↦ eLpNorm (↑↑(f n.1) - ↑↑(f n.2)) p μ) atTop (𝓝 0)

Tactic:
dist_def

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
ι : Type u_4
inst✝¹ : Nonempty ι
inst✝ : SemilatticeSup ι
hp : Fact (1 ≤ p)
f : ι → ↥(Lp E p μ)
⊢ Tendsto (fun n ↦ (eLpNorm (↑↑(f n.1) - ↑↑(f n.2)) p μ).toReal) atTop (𝓝 0) ↔
    Tendsto (fun n ↦ eLpNorm (↑↑(f n.1) - ↑↑(f n.2)) p μ) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
hB1 : ∑' (i : ℕ), B1 i < ∞
f1 : ℕ → α → E := fun n ↦ ↑↑(f n)
N n m : ℕ
hf : dist (f n) (f m) < B N
hn : N ≤ n
hm : N ≤ m
⊢ eLpNorm (f1 n - f1 m) p μ < B1 N

Tactic:
dist_def

Goal state after:
α : Type u_1
m✝ : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
hB1 : ∑' (i : ℕ), B1 i < ∞
f1 : ℕ → α → E := fun n ↦ ↑↑(f n)
N n m : ℕ
hf : (eLpNorm (↑↑(f n) - ↑↑(f m)) p μ).toReal < B N
hn : N ≤ n
hm : N ≤ m
⊢ eLpNorm (f1 n - f1 m) p μ < B1 N
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
⊢ TendstoInMeasure μ f atTop g

Tactic:
refine TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm ?_

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hg : AEStronglyMeasurable g μ
⊢ TendstoInMeasure μ (fun i ↦ AEStronglyMeasurable.mk (f i) ⋯) atTop (AEStronglyMeasurable.mk g hg)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : q = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ < ∞

pp: case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ < ∞

Tactic:
· simp only [q_zero, ENNReal.toReal_zero]
    by_cases p_zero : p = 0
    · simp [p_zero]
    rw [ENNReal.div_zero p_zero]
    simpa only [ENNReal.rpow_zero, eLpNorm_exponent_top] using (memLp_top_const_enorm (by simp)).2

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ < ∞
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h0
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
ε : ℝ≥0∞
hε : 0 < ε
k : ℕ
h_k : 2⁻¹ ^ k < ε
⊢ 2 ≠ 0

pp: case ht
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
ε : ℝ≥0∞
hε : 0 < ε
k : ℕ
h_k : 2⁻¹ ^ k < ε
⊢ 2 ≠ ∞

Tactic:
· positivity

Goal state after:
case ht
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
ε : ℝ≥0∞
hε : 0 < ε
k : ℕ
h_k : 2⁻¹ ^ k < ε
⊢ 2 ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
f f' : ↥(Lp E p μ)
⊢ ‖hg.compLp g0 f - hg.compLp g0 f'‖ ≤ ↑c * ‖f - f'‖

Tactic:
apply Lp.norm_le_mul_norm_of_ae_le_mul
  filter_upwards [hg.coeFn_compLp g0 f, hg.coeFn_compLp g0 f',
    Lp.coeFn_sub (hg.compLp g0 f) (hg.compLp g0 f'), Lp.coeFn_sub f f'] with a ha1 ha2 ha3 ha4
  simp only [ha1, ha2, ha3, ha4, ← dist_eq_norm, Pi.sub_apply, Function.comp_apply]
  exact hg.dist_le_mul (f a) (f' a)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
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 ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
⊢ (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i

Tactic:
by
  have hp_pos : 0 < p := zero_lt_one.trans_le hp1
  suffices h_pow : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ≤ (∑' i, B i) ^ p by
      rwa [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]
  have h_tsum_1 :
    ∀ g : ℕ → ℝ≥0∞, ∑' i, g i = atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), g i := by
    intro g
    rw [ENNReal.tsum_eq_liminf_sum_nat, ← liminf_nat_add _ 1]
  simp_rw [h_tsum_1 _]
  rw [← h_tsum_1]
  have h_liminf_pow :
    ∫⁻ a, (atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ =
      ∫⁻ a, atTop.liminf fun n => (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ 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
  rw [h_liminf_pow]
  refine (lintegral_liminf_le' fun n ↦ ?_).trans <| liminf_le_of_frequently_le' <| .of_forall h
  exact ((Finset.range _).aemeasurable_fun_sum fun i _ ↦ ((hf _).sub (hf i)).enorm).pow_const _

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
⊢ ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞

Tactic:
refine ae_lt_top' (AEMeasurable.pow_const ?_ _) h_integral.ne

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
⊢ AEMeasurable (fun x ↦ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) μ
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
l : Filter ι
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) l (𝓝 0)
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
⊢ 1 / ε ^ p.toReal ≠ ∞

Tactic:
by simp [hε.ne']

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
hg : AEStronglyMeasurable g μ
l : Filter ι
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) l (𝓝 0)
⊢ TendstoInMeasure μ f l g

Tactic:
refine TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm ?_

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
hg : AEStronglyMeasurable g μ
l : Filter ι
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) l (𝓝 0)
⊢ TendstoInMeasure μ (fun i ↦ AEStronglyMeasurable.mk (f i) ⋯) l (AEStronglyMeasurable.mk g hg)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : SeminormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) ∞ μ) l (𝓝 0)
δ : ℝ≥0∞
hδ : 0 < δ
hδ_top : δ ≠ ∞
⊢ Tendsto (fun i ↦ μ {x | δ ≤ edist (f i x) (g x)}) l (𝓝 0)

Tactic:
simp only [eLpNorm_exponent_top, eLpNormEssSup] at hfg

Goal state after:
α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : SeminormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
δ : ℝ≥0∞
hδ : 0 < δ
hδ_top : δ ≠ ∞
hfg : Tendsto (fun n ↦ essSup (fun x ↦ ‖(f n - g) x‖ₑ) μ) l (𝓝 0)
⊢ Tendsto (fun i ↦ μ {x | δ ≤ edist (f i x) (g x)}) l (𝓝 0)
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case neg
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : NormedAddCommGroup E
l : Filter ι
hp_ne_zero : p ≠ 0
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
hg : AEStronglyMeasurable g μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) l (𝓝 0)
hp_ne_top : ¬p = ∞
⊢ TendstoInMeasure μ f l g

Tactic:
exact tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top hp_ne_zero hp_ne_top hf hg hfg

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: case inr
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
hne : i ≠ i₀
⊢ update y i₀ ((a / ∏ i ∈ I, c i) • y i₀) i ∈ t i

Tactic:
simp_rw [update_of_ne hne, y, I.piecewise_eq_of_mem _ _ hi]

Goal state after:
case inr
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
hne : i ≠ i₀
⊢ c i • x i ∈ t i
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: case inr
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
hne : i ≠ i₀
⊢ c i • x i ∈ t i

Tactic:
exact hc _ _ le_rfl _ hx

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
a✝ : Nontrivial R
⊢ ‖↑x⁻¹‖⁻¹ > 0

Tactic:
by cancel_denoms

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: case pos
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : f x = 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Tactic:
rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps

Goal state before:
pp: case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ Tendsto (fun x_1 ↦ f (x_1.1 - x.1, x_1.2)) (𝓝 x) (𝓝 0)

Tactic:
← isLittleO_one_iff ℝ,

Goal state after:
case hf
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : Semiring 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : SeminormedAddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : Module 𝕜 G
f : E × F → G
h : IsBoundedBilinearMap 𝕜 f
x : E × F
⊢ (fun x_1 ↦ f (x_1.1 - x.1, x_1.2)) =o[𝓝 x] fun _x ↦ 1
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : NullMeasurableSet s μ
h₁ : μ s ≠ ∞
h₂ : μ t ≠ ∞
⊢ μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t

Tactic:
rw [union_comm, inter_comm, measureReal_union_add_inter₀ hs h₂ h₁, add_comm]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : Finite ι
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : TopologicalSpace F
inst✝² : AddCommGroup F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : Module 𝕜 F
⊢ Continuous[instTopologicalSpaceProd, inst✝³] fun fx ↦ fx.1 fx.2

Tactic:
cases nonempty_fintype ι

Goal state after:
case intro
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : Finite ι
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : TopologicalSpace F
inst✝² : AddCommGroup F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : Module 𝕜 F
val✝ : Fintype ι
⊢ Continuous[instTopologicalSpaceProd, inst✝³] fun fx ↦ fx.1 fx.2