l3lab/ntp-mathlib huggingface dataset:
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hfd : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z
hg : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
⊢ False

Next tactic:
specialize hfd c hc d hd
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; specialize hfd c hc d hd
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hg : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
⊢ False

Next tactic:
replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; specialize hfd c hc d hd
  replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
⊢ False

Next tactic:
have hg_ae : ∀ᵐ x ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖ :=
    (ae_restrict_mem measurableSet_uIoc).mono hg
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; specialize hfd c hc d hd
  replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩
  have hg_ae : ∀ᵐ x ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖ :=
    (ae_restrict_mem measurableSet_uIoc).mono hg
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂Measure.restrict volume (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
⊢ False

Next tactic:
have hsub' : Ι c d ⊆ k := Subset.trans Ioc_subset_Icc_self hsub
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; specialize hfd c hc d hd
  replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩
  have hg_ae : ∀ᵐ x ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖ :=
    (ae_restrict_mem measurableSet_uIoc).mono hg
  have hsub' : Ι c d ⊆ k := Subset.trans Ioc_subset_Icc_self hsub
  have hfi : IntervalIntegrable (deriv f) volume c d := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂Measure.restrict volume (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
⊢ IntervalIntegrable (deriv f) volume c d

Next tactic:
rw [intervalIntegrable_iff]
/-- If `f` is eventually differentiable along a nontrivial filter `l : Filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any set `k` belonging to `l`.
Auxiliary version assuming that `E` is complete. -/
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  intro hgi
  obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ :
    ∃ (C : ℝ) (_ : 0 ≤ C), ∃ s ∈ l, (∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k) ∧
      (∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z) ∧
        ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖ := by
    rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩
    have h : ∀ᶠ x : ℝ × ℝ in l.prod l,
        ∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
      (tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets
    rcases mem_prod_self_iff.1 h with ⟨s, hsl, hs⟩
    simp only [prod_subset_iff, mem_setOf_eq] at hs
    exact ⟨C, C₀, s, hsl, fun x hx y hy z hz => (hs x hx y hy z hz).2, fun x hx y hy z hz =>
      (hs x hx y hy z hz).1.1, fun x hx y hy z hz => (hs x hx y hy z hz).1.2⟩
  replace hgi : IntegrableOn (fun x ↦ C * ‖g x‖) k := by exact hgi.norm.smul C
  obtain ⟨c, hc, d, hd, hlt⟩ : ∃ c ∈ s, ∃ d ∈ s, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f d‖ := by
    rcases Filter.nonempty_of_mem hsl with ⟨c, hc⟩
    have : ∀ᶠ x in l, (‖f c‖ + ∫ y in k, C * ‖g y‖) < ‖f x‖ :=
      hf.eventually (eventually_gt_atTop _)
    exact ⟨c, hc, (this.and hsl).exists.imp fun d hd => ⟨hd.2, hd.1⟩⟩
  specialize hsub c hc d hd; specialize hfd c hc d hd
  replace hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖ :=
    fun z hz => hg c hc d hd z ⟨hz.1.le, hz.2⟩
  have hg_ae : ∀ᵐ x ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖ :=
    (ae_restrict_mem measurableSet_uIoc).mono hg
  have hsub' : Ι c d ⊆ k := Subset.trans Ioc_subset_Icc_self hsub
  have hfi : IntervalIntegrable (deriv f) volume c d := by
    rw [intervalIntegrable_iff]
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
C : ℝ
hC₀ : 0 ≤ C
s : Set ℝ
hsl : s ∈ l
hgi : IntegrableOn (fun x => C * ‖g x‖) k
c : ℝ
hc : c ∈ s
d : ℝ
hd : d ∈ s
hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖
hsub : [[c, d]] ⊆ k
hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z
hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖
hg_ae : ∀ᵐ (x : ℝ) ∂Measure.restrict volume (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖
hsub' : Ι c d ⊆ k
⊢ IntegrableOn (deriv f) (Ι c d)

Next tactic:
have : IntegrableOn (fun x ↦ C * ‖g x‖) (Ι c d) := IntegrableOn.mono hgi hsub' le_rfl
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
⊢ ¬IntegrableOn g k

Next tactic:
let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
⊢ ¬IntegrableOn g k

Next tactic:
let f' := a ∘ f
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
  let f' := a ∘ f
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
⊢ ¬IntegrableOn g k

Next tactic:
have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
    filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
  let f' := a ∘ f
  have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
    

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
⊢ ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x

Next tactic:
filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
  let f' := a ∘ f
  have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
    filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
  

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
⊢ ¬IntegrableOn g k

Next tactic:
have h'f : Tendsto (fun x => ‖f' x‖) l atTop := hf.congr (fun x ↦ by simp)
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k := by
  let a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
  let f' := a ∘ f
  have h'd : ∀ᶠ x in l, DifferentiableAt ℝ f' x := by
    filter_upwards [hd] with x hx using a.toContinuousLinearMap.differentiableAt.comp x hx
  have h'f : Tendsto (fun x => ‖f' x‖) l atTop := hf.congr (fun x ↦ by 

Proof state:
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : NeBot l
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x => ‖f x‖) l atTop
hfg : deriv f =O[l] g
a : E →ₗᵢ[ℝ] UniformSpace.Completion E := UniformSpace.Completion.toComplₗᵢ
f' : ℝ → UniformSpace.Completion E := ⇑a ∘ f
h'd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f' x
x : ℝ
⊢ ‖f x‖ = ‖f' x‖

Next tactic:
simp