l3lab/ntp-mathlib huggingface dataset:
theorem hasFDerivWithinAt_insert {y : E} :
    HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x := by
  rcases eq_or_ne x y with (rfl | h)
  · simp_rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO]
    apply Asymptotics.isLittleO_insert
    

Proof state:
case inl.h
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
⊢ f x - f x - f' (x - x) = 0

Next tactic:
simp only [sub_self, map_zero]
theorem hasFDerivWithinAt_insert {y : E} :
    HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x := by
  rcases eq_or_ne x y with (rfl | h)
  · simp_rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO]
    apply Asymptotics.isLittleO_insert
    simp only [sub_self, map_zero]
  

Proof state:
case inr
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
y : E
h : x ≠ y
⊢ HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x

Next tactic:
refine' ⟨fun h => h.mono <| subset_insert y s, fun hf => hf.mono_of_mem _⟩
theorem hasFDerivWithinAt_insert {y : E} :
    HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x := by
  rcases eq_or_ne x y with (rfl | h)
  · simp_rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO]
    apply Asymptotics.isLittleO_insert
    simp only [sub_self, map_zero]
  refine' ⟨fun h => h.mono <| subset_insert y s, fun hf => hf.mono_of_mem _⟩
  

Proof state:
case inr
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
y : E
h : x ≠ y
hf : HasFDerivWithinAt f f' s x
⊢ s ∈ 𝓝[insert y s] x

Next tactic:
simp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]
theorem hasFDerivWithinAt_diff_singleton (y : E) :
    HasFDerivWithinAt f f' (s \ {y}) x ↔ HasFDerivWithinAt f f' s x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
y : E
⊢ HasFDerivWithinAt f f' (s \ {y}) x ↔ HasFDerivWithinAt f f' s x

Next tactic:
rw [← hasFDerivWithinAt_insert, insert_diff_singleton, hasFDerivWithinAt_insert]
protected theorem HasStrictFDerivAt.hasFDerivAt (hf : HasStrictFDerivAt f f' x) :
    HasFDerivAt f f' x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
⊢ HasFDerivAt f f' x

Next tactic:
rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff]
protected theorem HasStrictFDerivAt.hasFDerivAt (hf : HasStrictFDerivAt f f' x) :
    HasFDerivAt f f' x := by
  rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff]
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝 x, ‖f x_1 - f x - f' (x_1 - x)‖ ≤ c * ‖x_1 - x‖

Next tactic:
exact fun c hc => tendsto_id.prod_mk_nhds tendsto_const_nhds (isLittleO_iff.1 hf hc)
/-- If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. -/
theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
    (K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, LipschitzOnWith K f s := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
K : ℝ≥0
hK : ‖f'‖₊ < K
⊢ ∃ s ∈ 𝓝 x, LipschitzOnWith K f s

Next tactic:
have := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK
/-- If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. -/
theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
    (K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, LipschitzOnWith K f s := by
  have := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
K : ℝ≥0
hK : ‖f'‖₊ < K
this :
  IsBigOWith ((fun a => ↑a) K) (𝓝 (x, x)) (fun x => f x.1 - f x.2 - f' (x.1 - x.2) + f' (x.1 - x.2)) fun p => p.1 - p.2
⊢ ∃ s ∈ 𝓝 x, LipschitzOnWith K f s

Next tactic:
simp only [sub_add_cancel, IsBigOWith] at this
/-- If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. -/
theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
    (K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, LipschitzOnWith K f s := by
  have := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK
  simp only [sub_add_cancel, IsBigOWith] at this
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
K : ℝ≥0
hK : ‖f'‖₊ < K
this : ∀ᶠ (x : E × E) in 𝓝 (x, x), ‖f x.1 - f x.2‖ ≤ ↑K * ‖x.1 - x.2‖
⊢ ∃ s ∈ 𝓝 x, LipschitzOnWith K f s

Next tactic:
rcases exists_nhds_square this with ⟨U, Uo, xU, hU⟩
/-- If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. -/
theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
    (K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, LipschitzOnWith K f s := by
  have := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK
  simp only [sub_add_cancel, IsBigOWith] at this
  rcases exists_nhds_square this with ⟨U, Uo, xU, hU⟩
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasStrictFDerivAt f f' x
K : ℝ≥0
hK : ‖f'‖₊ < K
this : ∀ᶠ (x : E × E) in 𝓝 (x, x), ‖f x.1 - f x.2‖ ≤ ↑K * ‖x.1 - x.2‖
U : Set E
Uo : IsOpen U
xU : x ∈ U
hU : U ×ˢ U ⊆ {x | (fun x => ‖f x.1 - f x.2‖ ≤ ↑K * ‖x.1 - x.2‖) x}
⊢ ∃ s ∈ 𝓝 x, LipschitzOnWith K f s

Next tactic:
exact
    ⟨U, Uo.mem_nhds xU, lipschitzOnWith_iff_norm_sub_le.2 fun x hx y hy => hU (mk_mem_prod hx hy)⟩
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
⊢ Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v))

Next tactic:
refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
⊢ Tendsto (fun n => c n • (c n)⁻¹ • v) l (𝓝 v)

Next tactic:
intro U hU
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
  intro U hU
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
U : Set E
hU : U ∈ 𝓝 v
⊢ U ∈ map (fun n => c n • (c n)⁻¹ • v) l

Next tactic:
refine' (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => _
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
  intro U hU
  refine' (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => _
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
U : Set E
hU : U ∈ 𝓝 v
y : α
hy : c y ≠ 0
⊢ (fun n => c n • (c n)⁻¹ • v) y ∈ U

Next tactic:
convert mem_of_mem_nhds hU
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
  intro U hU
  refine' (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => _
  convert mem_of_mem_nhds hU
  

Proof state:
case h.e'_4
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
U : Set E
hU : U ∈ 𝓝 v
y : α
hy : c y ≠ 0
⊢ (fun n => c n • (c n)⁻¹ • v) y = v

Next tactic:
dsimp only
/-- Directional derivative agrees with `HasFDeriv`. -/
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) := by
  refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _
  intro U hU
  refine' (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => _
  convert mem_of_mem_nhds hU
  dsimp only
  

Proof state:
case h.e'_4
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hf : HasFDerivAt f f' x
v : E
α : Type u_6
c : α → 𝕜
l : Filter α
hc : Tendsto (fun n => ‖c n‖) l atTop
U : Set E
hU : U ∈ 𝓝 v
y : α
hy : c y ≠ 0
⊢ c y • (c y)⁻¹ • v = v

Next tactic:
rw [← mul_smul, mul_inv_cancel hy, one_smul]
theorem HasFDerivAt.unique (h₀ : HasFDerivAt f f₀' x) (h₁ : HasFDerivAt f f₁' x) : f₀' = f₁' := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h₀ : HasFDerivAt f f₀' x
h₁ : HasFDerivAt f f₁' x
⊢ f₀' = f₁'

Next tactic:
rw [← hasFDerivWithinAt_univ] at h₀ h₁
theorem HasFDerivAt.unique (h₀ : HasFDerivAt f f₀' x) (h₁ : HasFDerivAt f f₁' x) : f₀' = f₁' := by
  rw [← hasFDerivWithinAt_univ] at h₀ h₁
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h₀ : HasFDerivWithinAt f f₀' univ x
h₁ : HasFDerivWithinAt f f₁' univ x
⊢ f₀' = f₁'

Next tactic:
exact uniqueDiffWithinAt_univ.eq h₀ h₁
theorem hasFDerivWithinAt_inter' (h : t ∈ 𝓝[s] x) :
    HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : t ∈ 𝓝[s] x
⊢ HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x

Next tactic:
simp [HasFDerivWithinAt, nhdsWithin_restrict'' s h]
theorem hasFDerivWithinAt_inter (h : t ∈ 𝓝 x) :
    HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : t ∈ 𝓝 x
⊢ HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x

Next tactic:
simp [HasFDerivWithinAt, nhdsWithin_restrict' s h]
theorem HasFDerivWithinAt.union (hs : HasFDerivWithinAt f f' s x)
    (ht : HasFDerivWithinAt f f' t x) : HasFDerivWithinAt f f' (s ∪ t) x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hs : HasFDerivWithinAt f f' s x
ht : HasFDerivWithinAt f f' t x
⊢ HasFDerivWithinAt f f' (s ∪ t) x

Next tactic:
simp only [HasFDerivWithinAt, nhdsWithin_union]
theorem HasFDerivWithinAt.union (hs : HasFDerivWithinAt f f' s x)
    (ht : HasFDerivWithinAt f f' t x) : HasFDerivWithinAt f f' (s ∪ t) x := by
  simp only [HasFDerivWithinAt, nhdsWithin_union]
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hs : HasFDerivWithinAt f f' s x
ht : HasFDerivWithinAt f f' t x
⊢ HasFDerivAtFilter f f' x (𝓝[s] x ⊔ 𝓝[t] x)

Next tactic:
exact .of_isLittleO <| hs.isLittleO.sup ht.isLittleO
theorem HasFDerivWithinAt.hasFDerivAt (h : HasFDerivWithinAt f f' s x) (hs : s ∈ 𝓝 x) :
    HasFDerivAt f f' x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : HasFDerivWithinAt f f' s x
hs : s ∈ 𝓝 x
⊢ HasFDerivAt f f' x

Next tactic:
rwa [← univ_inter s, hasFDerivWithinAt_inter hs, hasFDerivWithinAt_univ] at h
/-- If `x` is isolated in `s`, then `f` has any derivative at `x` within `s`,
as this statement is empty. -/
theorem HasFDerivWithinAt.of_nhdsWithin_eq_bot (h : 𝓝[s\{x}] x = ⊥) :
    HasFDerivWithinAt f f' s x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : 𝓝[s \ {x}] x = ⊥
⊢ HasFDerivWithinAt f f' s x

Next tactic:
rw [← hasFDerivWithinAt_diff_singleton x, HasFDerivWithinAt, h, hasFDerivAtFilter_iff_isLittleO]
/-- If `x` is isolated in `s`, then `f` has any derivative at `x` within `s`,
as this statement is empty. -/
theorem HasFDerivWithinAt.of_nhdsWithin_eq_bot (h : 𝓝[s\{x}] x = ⊥) :
    HasFDerivWithinAt f f' s x := by
  rw [← hasFDerivWithinAt_diff_singleton x, HasFDerivWithinAt, h, hasFDerivAtFilter_iff_isLittleO]
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : 𝓝[s \ {x}] x = ⊥
⊢ (fun x' => f x' - f x - f' (x' - x)) =o[⊥] fun x' => x' - x

Next tactic:
apply isLittleO_bot
/-- If `x` is not in the closure of `s`, then `f` has any derivative at `x` within `s`,
as this statement is empty. -/
theorem hasFDerivWithinAt_of_nmem_closure (h : x ∉ closure s) : HasFDerivWithinAt f f' s x :=
  .of_nhdsWithin_eq_bot <| eq_bot_mono (nhdsWithin_mono _ (diff_subset _ _)) <| by
    

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : x ∉ closure s
⊢ 𝓝[s] x = ⊥

Next tactic:
rwa [mem_closure_iff_nhdsWithin_neBot, not_neBot] at h
theorem DifferentiableWithinAt.hasFDerivWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x := by
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : DifferentiableWithinAt 𝕜 f s x
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x

Next tactic:
by_cases H : 𝓝[s \ {x}] x = ⊥
theorem DifferentiableWithinAt.hasFDerivWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x := by
  by_cases H : 𝓝[s \ {x}] x = ⊥
  · 

Proof state:
case pos
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : DifferentiableWithinAt 𝕜 f s x
H : 𝓝[s \ {x}] x = ⊥
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x

Next tactic:
exact .of_nhdsWithin_eq_bot H