Theorem: not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
File: Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean

tactic: have : l ≤ 𝓝[[[a, b]] \ {c}] c := le_inf hle (le_principal_iff.2 hmem)
state_before: case intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
l : Filter ℝ
hl : TendstoIxxClass Icc l l
hl' : NeBot l
hle : l ≤ 𝓝 c
hmem : [[a, b]] \ {c} ∈ l
⊢ ¬IntervalIntegrable g volume a b

tactic: exact not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter l
  (mem_of_superset hmem (diff_subset _ _)) (h_deriv.filter_mono this) (h_infty.mono_left this)
  (hg.mono this)
state_before: case intro.intro.intro.intro
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
l : Filter ℝ
hl : TendstoIxxClass Icc l l
hl' : NeBot l
hle : l ≤ 𝓝 c
hmem : [[a, b]] \ {c} ∈ l
this : l ≤ 𝓝[[[a, b]] \ {c}] c
⊢ ¬IntervalIntegrable g volume a b

tactic: cases' (min_lt_max.2 hne).lt_or_lt c with hlt hlt
state_before: E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
⊢ ∃ l, TendstoIxxClass Icc l l ∧ NeBot l ∧ l ≤ 𝓝 c ∧ [[a, b]] \ {c} ∈ l

tactic: refine' ⟨𝓝[<] c, inferInstance, inferInstance, inf_le_left, _⟩
state_before: case inl
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : min a b < c
⊢ ∃ l, TendstoIxxClass Icc l l ∧ NeBot l ∧ l ≤ 𝓝 c ∧ [[a, b]] \ {c} ∈ l

tactic: rw [← Iic_diff_right]
state_before: case inl
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : min a b < c
⊢ [[a, b]] \ {c} ∈ 𝓝[Iio c] c

tactic: exact diff_mem_nhdsWithin_diff (Icc_mem_nhdsWithin_Iic ⟨hlt, hc.2⟩) _
state_before: case inl
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : min a b < c
⊢ [[a, b]] \ {c} ∈ 𝓝[Iic c \ {c}] c

tactic: refine' ⟨𝓝[>] c, inferInstance, inferInstance, inf_le_left, _⟩
state_before: case inr
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : c < max a b
⊢ ∃ l, TendstoIxxClass Icc l l ∧ NeBot l ∧ l ≤ 𝓝 c ∧ [[a, b]] \ {c} ∈ l

tactic: rw [← Ici_diff_left]
state_before: case inr
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : c < max a b
⊢ [[a, b]] \ {c} ∈ 𝓝[Ioi c] c

tactic: exact diff_mem_nhdsWithin_diff (Icc_mem_nhdsWithin_Ici ⟨hc.1, hlt⟩) _
state_before: case inr
E : Type u_1
F : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : CompleteSpace E
inst✝ : NormedAddCommGroup F
f : ℝ → E
g : ℝ → F
a b c : ℝ
hne : a ≠ b
hc : c ∈ [[a, b]]
h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x
h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop
hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g
hlt : c < max a b
⊢ [[a, b]] \ {c} ∈ 𝓝[Ici c \ {c}] c