l3lab/ntp-mathlib huggingface dataset:
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · 

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N
hp_top : p = ⊤
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)

Next tactic:
simp_rw [hp_top] at *
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) ⊤ μ < B N
hp_top : True
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)

Next tactic:
have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) ⊤ μ < B N
hp_top : True
⊢ ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N

Next tactic:
simp_rw [ae_all_iff]
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) ⊤ μ < B N
hp_top : True
⊢ ∀ (i i_1 i_2 : ℕ), i ≤ i_1 → i ≤ i_2 → ∀ᵐ (a : α) ∂μ, ↑‖(f i_1 - f i_2) a‖₊ < B i

Next tactic:
exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) ⊤ μ < B N
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)

Next tactic:
simp_rw [snorm_exponent_top, snormEssSup] at h_cau
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)

Next tactic:
refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    

Proof state:
case pos
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
⊢ CauchySeq fun n => f n x

Next tactic:
refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · 

Proof state:
case pos.refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
⊢ ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n x) (f m x) ≤ (fun n => ENNReal.toReal (B n)) N

Next tactic:
intro n m N hnN hmN
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      

Proof state:
case pos.refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m✝ m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
n m N : ℕ
hnN : N ≤ n
hmN : N ≤ m
⊢ dist (f n x) (f m x) ≤ (fun n => ENNReal.toReal (B n)) N

Next tactic:
specialize hx N n m hnN hmN
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      

Proof state:
case pos.refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m✝ m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
n m N : ℕ
hnN : N ≤ n
hmN : N ≤ m
hx : ↑‖(f n - f m) x‖₊ < B N
⊢ dist (f n x) (f m x) ≤ (fun n => ENNReal.toReal (B n)) N

Next tactic:
rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      

Proof state:
case pos.refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m✝ m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
n m N : ℕ
hnN : N ≤ n
hmN : N ≤ m
hx : ↑‖(f n - f m) x‖₊ < B N
⊢ ENNReal.ofReal ‖f n x - f m x‖ ≤ B N

Next tactic:
rw [← ofReal_norm_eq_coe_nnnorm] at hx
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      rw [← ofReal_norm_eq_coe_nnnorm] at hx
      

Proof state:
case pos.refine'_1
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m✝ m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
n m N : ℕ
hnN : N ≤ n
hmN : N ≤ m
hx : ENNReal.ofReal ‖(f n - f m) x‖ < B N
⊢ ENNReal.ofReal ‖f n x - f m x‖ ≤ B N

Next tactic:
exact hx.le
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      rw [← ofReal_norm_eq_coe_nnnorm] at hx
      exact hx.le
    · 

Proof state:
case pos.refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
⊢ Tendsto (fun n => ENNReal.toReal (B n)) atTop (𝓝 0)

Next tactic:
rw [← ENNReal.zero_toReal]
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      rw [← ofReal_norm_eq_coe_nnnorm] at hx
      exact hx.le
    · rw [← ENNReal.zero_toReal]
      

Proof state:
case pos.refine'_2
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
hp : 1 ≤ ⊤
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → essSup (fun x => ↑‖(f n - f m) x‖₊) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ↑‖(f n - f m) x‖₊ < B N
⊢ Tendsto (fun n => ENNReal.toReal (B n)) atTop (𝓝 (ENNReal.toReal 0))

Next tactic:
exact
        Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)
          (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      rw [← ofReal_norm_eq_coe_nnnorm] at hx
      exact hx.le
    · rw [← ENNReal.zero_toReal]
      exact
        Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)
          (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)
  

Proof state:
case neg
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N
hp_top : ¬p = ⊤
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)

Next tactic:
have hp1 : 1 ≤ p.toReal := by
    rw [← ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]
    exact hp
theorem ae_tendsto_of_cauchy_snorm [CompleteSpace E] {f : ℕ → α → E}
    (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N) :
    ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
  by_cases hp_top : p = ∞
  · simp_rw [hp_top] at *
    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
      simp_rw [ae_all_iff]
      exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
    simp_rw [snorm_exponent_top, snormEssSup] at h_cau
    refine' h_cau_ae.mono fun x hx => cauchySeq_tendsto_of_complete _
    refine' cauchySeq_of_le_tendsto_0 (fun n => (B n).toReal) _ _
    · intro n m N hnN hmN
      specialize hx N n m hnN hmN
      rw [dist_eq_norm, ← ENNReal.toReal_ofReal (norm_nonneg _),
        ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.ne_top_of_tsum_ne_top hB N)]
      rw [← ofReal_norm_eq_coe_nnnorm] at hx
      exact hx.le
    · rw [← ENNReal.zero_toReal]
      exact
        Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)
          (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)
  have hp1 : 1 ≤ p.toReal := by
    

Proof state:
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → snorm (f n - f m) p μ < B N
hp_top : ¬p = ⊤
⊢ 1 ≤ ENNReal.toReal p

Next tactic:
rw [← ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]