mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.ReImTopology

Goal state before:
pp: s t : Set ℝ
⊢ closure (s ×ℂ t) = closure s ×ℂ closure t

Tactic:
by
  simpa only [← preimage_eq_preimage equivRealProdCLM.symm.toHomeomorph.surjective,
    equivRealProdCLM.symm.toHomeomorph.preimage_closure] using! @closure_prod_eq _ _ _ _ s t

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
f' : F
x : 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
t : Set F
hl : HasFDerivWithinAt l l' t y
hf : HasDerivAt f f' x
ht : ∀ᶠ (x' : 𝕜) in 𝓝 x, f x' ∈ t
hy : y = f x
⊢ HasDerivAt (l ∘ f) (l' f') x

Tactic:
by
  subst y; exact hl.comp_hasDerivAt x hf ht

Goal state after:
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
s : Set E
f : 𝕜 → E
hfU : AnalyticOnNhd 𝕜 f U
h₂f : ∀ x ∈ U, ¬EventuallyConst f (𝓝 x)
hs : ∀ x ∈ f '' U, s ∪ (f '' U)ᶜ ∈ 𝓝[≠] x
x : 𝕜
⊢ x ∈ U → f ⁻¹' s ∪ Uᶜ ∈ 𝓝[≠] x

Tactic:
hx

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
s : Set E
f : 𝕜 → E
hfU : AnalyticOnNhd 𝕜 f U
h₂f : ∀ x ∈ U, ¬EventuallyConst f (𝓝 x)
hs : ∀ x ∈ f '' U, s ∪ (f '' U)ᶜ ∈ 𝓝[≠] x
x : 𝕜
hx : x ∈ U
⊢ f ⁻¹' s ∪ Uᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Calculus.ContDiff.RCLike

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
hs : Convex ℝ s
hf : ContDiffOn ℝ 1 f s
x : E
⊢ x ∈ s → ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Tactic:
hx

Goal state after:
E : Type u_4
F : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
hs : Convex ℝ s
hf : ContDiffOn ℝ 1 f s
x : E
hx : x ∈ s
⊢ ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: case hB'
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x✝ : ℝ
hx : x✝ ∈ Icc a b
r : ℝ
hr : r > 0
x : ℝ
⊢ x ∈ Ico a b → HasDerivWithinAt (fun x ↦ B x + r * (x - a)) (?m.110 x) (Ici x) x

Tactic:
hx

Goal state after:
case hB'
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x✝ : ℝ
hx✝ : x✝ ∈ Icc a b
r : ℝ
hr : r > 0
x : ℝ
hx : x ∈ Ico a b
⊢ HasDerivWithinAt (fun x ↦ B x + r * (x - a)) (?m.110 x) (Ici x) x
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a)
x : ℝ
⊢ x ∈ Icc a b → f x ≤ B x

Tactic:
hx

Goal state after:
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a)
x : ℝ
hx : x ∈ Icc a b
⊢ f x ≤ B x
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
x : ℝ
⊢ x ∈ Ico a b → HasDerivWithinAt g (f' x) (Ici x) x

Tactic:
hx

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
f' : ℝ → E
C : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C
g : ℝ → E := fun x ↦ f x - f a
hg : ContinuousOn g (Icc a b)
x : ℝ
hx : x ∈ Ico a b
⊢ HasDerivWithinAt g (f' x) (Ici x) x
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
C : ℝ
hf : ∀ x ∈ Icc 0 1, HasDerivWithinAt f (f' x) (Icc 0 1) x
bound : ∀ x ∈ Ico 0 1, ‖f' x‖ ≤ C
⊢ ‖f 1 - f 0‖ ≤ C

Tactic:
simpa only [sub_zero, mul_one] using
    norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiff 𝕜 n f

Tactic:
rw [Subsingleton.elim f fun _ => 0]; exact contDiff_const

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
⊢ x ∈ ⇑g ⁻¹' s → ((p (g x) 0).compContinuousLinearMap fun x ↦ g.contLinear).curry0 = (f ∘ ⇑g) x

Tactic:
hx

Goal state after:
case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ ((p (g x) 0).compContinuousLinearMap fun x ↦ g.contLinear).curry0 = (f ∘ ⇑g) x
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ ((p (g x) 0) fun x ↦ g.contLinear 0) = (p (g x) 0) 0

Tactic:
map_zero

Goal state after:
case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ ((p (g x) 0) fun x ↦ 0) = (p (g x) 0) 0
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case fderivWithin
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
m : ℕ
hm : ↑m < n
x : G
⊢ x ∈ ⇑g ⁻¹' s →
    HasFDerivWithinAt (fun x ↦ (p (g x) m).compContinuousLinearMap fun x ↦ g.contLinear)
      ((p (g x) m.succ).compContinuousLinearMap fun x ↦ g.contLinear).curryLeft (⇑g ⁻¹' s) x

Tactic:
hx

Goal state after:
case fderivWithin
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
m : ℕ
hm : ↑m < n
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ HasFDerivWithinAt (fun x ↦ (p (g x) m).compContinuousLinearMap fun x ↦ g.contLinear)
    ((p (g x) m.succ).compContinuousLinearMap fun x ↦ g.contLinear).curryLeft (⇑g ⁻¹' s) x
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
x₀ : ℝ
n : ℕ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf' : f' =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n
h : (fun x ↦ f x - f x₀) =o[𝓝[s] x₀] fun x ↦ ‖x - x₀‖ ^ (n + 1)
⊢ (fun x ↦ f x - f x₀) =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ (n + 1)

Tactic:
rw [Asymptotics.isLittleO_iff] at h ⊢

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f f' : ℝ → E
x₀ : ℝ
n : ℕ
s : Set ℝ
hs : Convex ℝ s
hx₀s : x₀ ∈ s
hff' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf' : f' =o[𝓝[s] x₀] fun x ↦ (x - x₀) ^ n
h : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖f x - f x₀‖ ≤ c * ‖‖x - x₀‖ ^ (n + 1)‖
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in 𝓝[s] x₀, ‖f x - f x₀‖ ≤ c * ‖(x - x₀) ^ (n + 1)‖
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
hf : HasFTaylorSeriesUpToOn n f p s
g : E → G
q : E → FormalMultilinearSeries 𝕜 E G
hg : HasFTaylorSeriesUpToOn n g q s
L : (m : ℕ) → (E [×m]→L[𝕜] F) × (E [×m]→L[𝕜] G) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] F × G :=
  fun m ↦ ContinuousMultilinearMap.prodL 𝕜 (fun x ↦ E) F G
x : E
⊢ x ∈ s → ((p x 0).prod (q x 0)).curry0 = (f x, g x)

Tactic:
hx

Goal state after:
case zero_eq
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
hf : HasFTaylorSeriesUpToOn n f p s
g : E → G
q : E → FormalMultilinearSeries 𝕜 E G
hg : HasFTaylorSeriesUpToOn n g q s
L : (m : ℕ) → (E [×m]→L[𝕜] F) × (E [×m]→L[𝕜] G) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] F × G :=
  fun m ↦ ContinuousMultilinearMap.prodL 𝕜 (fun x ↦ E) F G
x : E
hx : x ∈ s
⊢ ((p x 0).prod (q x 0)).curry0 = (f x, g x)
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case fderivWithin
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
hf : HasFTaylorSeriesUpToOn n f p s
g : E → G
q : E → FormalMultilinearSeries 𝕜 E G
hg : HasFTaylorSeriesUpToOn n g q s
L : (m : ℕ) → (E [×m]→L[𝕜] F) × (E [×m]→L[𝕜] G) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] F × G :=
  fun m ↦ ContinuousMultilinearMap.prodL 𝕜 (fun x ↦ E) F G
m : ℕ
hm : ↑m < n
x : E
⊢ x ∈ s → HasFDerivWithinAt (fun x ↦ (p x m).prod (q x m)) ((p x m.succ).prod (q x m.succ)).curryLeft s x

Tactic:
hx

Goal state after:
case fderivWithin
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ∞ω
hf : HasFTaylorSeriesUpToOn n f p s
g : E → G
q : E → FormalMultilinearSeries 𝕜 E G
hg : HasFTaylorSeriesUpToOn n g q s
L : (m : ℕ) → (E [×m]→L[𝕜] F) × (E [×m]→L[𝕜] G) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] F × G :=
  fun m ↦ ContinuousMultilinearMap.prodL 𝕜 (fun x ↦ E) F G
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
⊢ HasFDerivWithinAt (fun x ↦ (p x m).prod (q x m)) ((p x m.succ).prod (q x m.succ)).curryLeft s x
Module: Mathlib.Analysis.Calculus.FDeriv.OfCompLeft

Goal state before:
pp: case ho
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : E → F
f : F → G
h : E → G
f' : F →L[𝕜] G
h' : E →L[𝕜] G
f'symm : G →L[𝕜] F
lE : Filter (E × E)
lF : Filter (F × F)
hg : Tendsto (Prod.map g g) lE lF
hf : HasFDerivAtFilter f f' lF
hh : HasFDerivAtFilter h h' lE
hcomp : Prod.map (f ∘ g) (f ∘ g) =ᶠ[lE] Prod.map h h
hf'symm : Function.LeftInverse ⇑f'symm ⇑f'
⊢ (fun x ↦
      match x with
      | (x, y) => g x - g y - (f'symm ∘SL h') (x - y)) =O[𝕜;
    lE] fun x ↦
    match x with
    | (x, y) => f' (g x - g y) - h' (x - y)

Tactic:
refine f'symm.isBigOTVS_comp.congr_left ?_

Goal state after:
case ho
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : E → F
f : F → G
h : E → G
f' : F →L[𝕜] G
h' : E →L[𝕜] G
f'symm : G →L[𝕜] F
lE : Filter (E × E)
lF : Filter (F × F)
hg : Tendsto (Prod.map g g) lE lF
hf : HasFDerivAtFilter f f' lF
hh : HasFDerivAtFilter h h' lE
hcomp : Prod.map (f ∘ g) (f ∘ g) =ᶠ[lE] Prod.map h h
hf'symm : Function.LeftInverse ⇑f'symm ⇑f'
⊢ ∀ (x : E × E),
    (⇑f'symm ∘ fun x ↦
          match x with
          | (x, y) => f' (g x - g y) - h' (x - y))
        x =
      match x with
      | (x, y) => g x - g y - (f'symm ∘SL h') (x - y)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case e'_10
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
hv : v ∈ 𝓝[insert x u] x
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
⊢ (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) ((p' x).unshift (f x) 1)) = fun x ↦ (p' x 0).curry0

Tactic:
ext x y

Goal state after:
case e'_10
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x✝ : E
n : ℕ∞ω
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x✝ s] x✝
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x✝
v : Set E
hv : v ∈ 𝓝[insert x✝ u] x✝
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
x y : E
⊢ ((continuousMultilinearCurryFin1 𝕜 E F) ((p' x).unshift (f x) 1)) y = (p' x 0).curry0 y
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
j : Fin n
m : Fin c.length
r : Fin (c.partSize m)
hmr : c.emb m r = j
⊢ ∃ p, c.emb p.fst p.snd = j

Tactic:
exact ⟨⟨m, r⟩, hmr⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
f' : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn (↑n) f f' s
x : E
⊢ x ∈ s → ContDiffWithinAt 𝕜 (↑n) f s x

Tactic:
hx

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞
f' : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn (↑n) f f' s
x : E
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 (↑n) f s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
left✝ : u ⊆ insert x s
hd : ContDiffOn 𝕜 n f u
⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y

Tactic:
have : ∀ᶠ y : E in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u :=
    (eventually_eventually_nhdsWithin.2 hu).and hu

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
left✝ : u ⊆ insert x s
hd : ContDiffOn 𝕜 n f u
this : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u
⊢ ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
h : ContDiffOn 𝕜 (n + 1) f s
x : E
⊢ x ∈ s →
    ∃ u ∈ 𝓝[insert x s] x,
      (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u

Tactic:
hx

Goal state after:
case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
h : ContDiffOn 𝕜 (n + 1) f s
x : E
hx : x ∈ s
⊢ ∃ u ∈ 𝓝[insert x s] x,
    (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mpr
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
h :
  ∀ x ∈ s,
    ∃ u ∈ 𝓝[insert x s] x,
      (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
x : E
⊢ x ∈ s → ContDiffWithinAt 𝕜 (n + 1) f s x

Tactic:
hx

Goal state after:
case mpr
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
hn : n ≠ ∞
h :
  ∀ x ∈ s,
    ∃ u ∈ 𝓝[insert x s] x,
      (n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u
x : E
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 (n + 1) f s x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
⊢ x ∈ s → iteratedDerivWithin n f s x = iteratedDeriv n f x

Tactic:
hx

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ iteratedDerivWithin n f s x = iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case fderivWithin
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
⊢ x ∈ s → HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x

Tactic:
hx

Goal state after:
case fderivWithin
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
⊢ HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case cont
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
⊢ x ∈ s →
    ∃ t,
      IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
        x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)

Tactic:
hx

Goal state after:
case cont
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m ≤ n
x : E
hx : x ∈ s
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p s ∧ ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) s
x : E
⊢ x ∈ s → ContDiffWithinAt 𝕜 ω f s x

Tactic:
hx

Goal state after:
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p s ∧ ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) s
x : E
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 ω f s x