mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Analytic.Inverse

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E✝ : Type u_2
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace 𝕜 E✝
F✝ : Type u_3
inst✝¹ : NormedAddCommGroup F✝
inst✝ : NormedSpace 𝕜 F✝
f : OpenPartialHomeomorph E✝ F✝
a : E✝
i : E✝ ≃L[𝕜] F✝
h0 : a ∈ f.source
p : FormalMultilinearSeries 𝕜 E✝ F✝
h : HasFPowerSeriesAt (↑f) p a
hp : p 1 = (continuousMultilinearCurryFin1 𝕜 E✝ F✝).symm ↑i
A : HasFPowerSeriesAt (↑f.symm ∘ ↑f) ((p.leftInv i a).comp p) a
B : ∀ᶠ (y : E✝) in 𝓝 0, HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f (a + y) - ↑f a) (↑f.symm (↑f (a + y)))
C : ∀ᶠ (y : E✝) in 𝓝 a, HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f y - ↑f a) (↑f.symm (↑f y))
D : ∀ᶠ (y : E✝) in 𝓝 (↑f.symm (↑f a)), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f y - ↑f a) y
E : ∀ᶠ (z : F✝) in 𝓝 (↑f a), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f (↑f.symm z) - ↑f a) (↑f.symm z)
F : ∀ᶠ (z : F✝) in 𝓝 (↑f a), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ z - ↑f a) (↑f.symm z)
r : ℝ≥0∞
r_pos : r > 0
hr : Metric.eball (↑f a) r ⊆ {x | (fun z ↦ HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ z - ↑f a) (↑f.symm z)) x}
⊢ HasFPowerSeriesAt (↑f.symm) (p.leftInv i a) (↑f a)

Tactic:
refine ⟨min r (p.leftInv i a).radius, min_le_right _ _,
    lt_min r_pos (radius_leftInv_pos_of_radius_pos h.radius_pos hp), fun {y} hy ↦ ?_⟩

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E✝ : Type u_2
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace 𝕜 E✝
F✝ : Type u_3
inst✝¹ : NormedAddCommGroup F✝
inst✝ : NormedSpace 𝕜 F✝
f : OpenPartialHomeomorph E✝ F✝
a : E✝
i : E✝ ≃L[𝕜] F✝
h0 : a ∈ f.source
p : FormalMultilinearSeries 𝕜 E✝ F✝
h : HasFPowerSeriesAt (↑f) p a
hp : p 1 = (continuousMultilinearCurryFin1 𝕜 E✝ F✝).symm ↑i
A : HasFPowerSeriesAt (↑f.symm ∘ ↑f) ((p.leftInv i a).comp p) a
B : ∀ᶠ (y : E✝) in 𝓝 0, HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f (a + y) - ↑f a) (↑f.symm (↑f (a + y)))
C : ∀ᶠ (y : E✝) in 𝓝 a, HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f y - ↑f a) (↑f.symm (↑f y))
D : ∀ᶠ (y : E✝) in 𝓝 (↑f.symm (↑f a)), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f y - ↑f a) y
E : ∀ᶠ (z : F✝) in 𝓝 (↑f a), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ ↑f (↑f.symm z) - ↑f a) (↑f.symm z)
F : ∀ᶠ (z : F✝) in 𝓝 (↑f a), HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ z - ↑f a) (↑f.symm z)
r : ℝ≥0∞
r_pos : r > 0
hr : Metric.eball (↑f a) r ⊆ {x | (fun z ↦ HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ z - ↑f a) (↑f.symm z)) x}
y : F✝
hy : y ∈ Metric.eball 0 (min r (p.leftInv i a).radius)
⊢ HasSum (fun n ↦ (p.leftInv i a n) fun x ↦ y) (↑f.symm (↑f a + y))
Module: Mathlib.Analysis.Calculus.FDeriv.Add

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
R : Type u_4
inst✝⁴ : Monoid R
inst✝³ : DistribMulAction R F
inst✝² : SMulCommClass 𝕜 R F
inst✝¹ : ContinuousConstSMul R F
c : R
inst✝ : Invertible c
h : DifferentiableWithinAt 𝕜 (c • f) s x
⊢ f =ᶠ[nhdsWithin x s] ⅟c • c • f

Tactic:
filter_upwards with x using by simp

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
H : DifferentiableWithinAt 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
⊢ MapsTo (⇑iso.symm) s (⇑iso ⁻¹' s)

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
H : DifferentiableWithinAt 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
y : F
⊢ y ∈ s → iso.symm y ∈ ⇑iso ⁻¹' s
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
A : f = (f ∘ ⇑iso) ∘ ⇑iso.symm
B : f' = (f' ∘SL ↑iso) ∘SL ↑iso.symm
⊢ MapsTo (⇑iso.symm) s (⇑iso ⁻¹' s)

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
iso : E ≃L[𝕜] F
f : F → G
s : Set F
x : E
f' : F →L[𝕜] G
H : HasFDerivWithinAt (f ∘ ⇑iso) (f' ∘SL ↑iso) (⇑iso ⁻¹' s) (iso.symm (iso x))
A : f = (f ∘ ⇑iso) ∘ ⇑iso.symm
B : f' = (f' ∘SL ↑iso) ∘SL ↑iso.symm
y : F
⊢ y ∈ s → iso.symm y ∈ ⇑iso ⁻¹' s
Module: Mathlib.Analysis.Calculus.FDeriv.Equiv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
f' : E →L[𝕜] F
x : E
h : HasFDerivWithinAt f f' s x
⊢ MapsTo (⇑f') (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x))

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
f' : E →L[𝕜] F
x : E
h : HasFDerivWithinAt f f' s x
y : E
⊢ y ∈ tangentConeAt 𝕜 s x → f' y ∈ tangentConeAt 𝕜 (f '' s) (f x)
Module: Mathlib.Analysis.Normed.Module.Completion

Goal state before:
pp: case hp
𝕜 : Type u_1
E : Type u_2
A : Type u_3
inst✝ : SeminormedRing A
⊢ IsClosed {x | ‖x.1 * x.2‖ ≤ ‖x.1‖ * ‖x.2‖}

Tactic:
apply isClosed_le <;> fun_prop

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

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
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
⊢ HasFTaylorSeriesUpToOn N f p s → ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s

Tactic:
intro H n

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
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
H : HasFTaylorSeriesUpToOn N f p s
n : ℕ
⊢ HasFTaylorSeriesUpToOn (↑(n + k)) f p s
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

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
p : E → FormalMultilinearSeries 𝕜 E F
hN : ∞ ≤ N
k : ℕ
H : ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑(n + k)) f p s
⊢ ∀ (m : ℕ), ↑m < N → ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x

Tactic:
intro m _
      apply (H m.succ).fderivWithin m (by norm_cast; lia)

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
⊢ DifferentiableOn 𝕜 f (insert x s ∩ Metric.eball x r)

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
y : E
⊢ y ∈ insert x s ∩ Metric.eball x r → DifferentiableWithinAt 𝕜 f (insert x s ∩ Metric.eball x r) y
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
n : WithTop ℕ∞
h : AnalyticWithinAt 𝕜 f s x
g : E → F
fg : EqOn f g (insert x s)
hg : AnalyticAt 𝕜 g x
v : Set E
vx : v ∈ 𝓝 x
hv : AnalyticOnNhd 𝕜 g v
⊢ HasFTaylorSeriesUpToOn n f (ftaylorSeries 𝕜 g) (insert x s ∩ v)

Tactic:
suffices HasFTaylorSeriesUpToOn n g (ftaylorSeries 𝕜 g) (insert x s ∩ v) from
      this.congr (fun y hy ↦ fg hy.1)
    exact AnalyticOnNhd.hasFTaylorSeriesUpToOn _ (hv.mono Set.inter_subset_right)

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinOnBall f p s x r
f' : E →L[𝕜] F
y : E
hy : ↑‖y‖₊ < r
h'y : x + y ∈ insert x s
hf' : HasFDerivWithinAt f f' (insert x s) (x + y)
hu : UniqueDiffOn 𝕜 (insert x s)
F' : Type v := UniformSpace.Completion F
a : F →L[𝕜] F' := UniformSpace.Completion.toComplL
b : (E →L[𝕜] F) →ₗᵢ[𝕜] E →L[𝕜] F' := UniformSpace.Completion.toComplₗᵢ.postcomp
this : HasFPowerSeriesWithinOnBall (⇑a ∘ f) (a.compFormalMultilinearSeries p) s x r
Z :
  HasSum (fun n ↦ ((a.compFormalMultilinearSeries p).derivSeries n) fun x ↦ y)
    (fderivWithin 𝕜 (⇑a ∘ f) (insert x s) (x + y))
⊢ fderivWithin 𝕜 (⇑a ∘ f) (insert x s) (x + y) = a ∘SL f'

Tactic:
apply HasFDerivWithinAt.fderivWithin _ (hu _ h'y)

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinOnBall f p s x r
f' : E →L[𝕜] F
y : E
hy : ↑‖y‖₊ < r
h'y : x + y ∈ insert x s
hf' : HasFDerivWithinAt f f' (insert x s) (x + y)
hu : UniqueDiffOn 𝕜 (insert x s)
F' : Type v := ⋯
a : F →L[𝕜] F' := ⋯
b : (E →L[𝕜] F) →ₗᵢ[𝕜] E →L[𝕜] F' := ⋯
this : HasFPowerSeriesWithinOnBall (⇑a ∘ f) (a.compFormalMultilinearSeries p) s x r
Z :
  HasSum (fun n ↦ ((a.compFormalMultilinearSeries p).derivSeries n) fun x ↦ y)
    (fderivWithin 𝕜 (⇑a ∘ f) (insert x s) (x + y))
⊢ HasFDerivWithinAt (⇑a ∘ f) (a ∘SL f') (insert x s) (x + y)
Module: Mathlib.Analysis.Calculus.FDeriv.Pow

Goal state before:
pp: 𝕜 : Type u_1
𝔸 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
x : 𝔸
n : ℕ
⊢ fderiv 𝕜 (fun x ↦ x ^ n) x = ∑ i ∈ Finset.range n, x ^ (n.pred - i) •> ContinuousLinearMap.id 𝕜 𝔸 <• x ^ i

Tactic:
fderiv_fun_pow' n differentiableAt_fun_id,

Goal state after:
𝕜 : Type u_1
𝔸 : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
x : 𝔸
n : ℕ
⊢ ∑ i ∈ Finset.range n, x ^ (n.pred - i) •> fderiv 𝕜 (fun x ↦ x) x <• x ^ i =
    ∑ i ∈ Finset.range n, x ^ (n.pred - i) •> ContinuousLinearMap.id 𝕜 𝔸 <• x ^ i
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : AnalyticOn 𝕜 f s
hu : UniqueDiffOn 𝕜 s
n : ℕ
IH : AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 n f s) s
⊢ AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 (n + 1) f s) s

Tactic:
rw [iteratedFDerivWithin_succ_eq_comp_left]
    apply AnalyticOnNhd.comp_analyticOn _ (IH.fderivWithin hu) (mapsTo_univ _ _)
    apply LinearIsometryEquiv.analyticOnNhd

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (_root_.fderiv 𝕜 f) s

Tactic:
intro y

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
y : E
⊢ y ∈ s → CPolynomialAt 𝕜 (_root_.fderiv 𝕜 f) y
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (_root_.fderiv 𝕜 f) s

Tactic:
intro y hy
  rcases h y hy with ⟨p, r, n, hp⟩
  exact hp.fderiv'.cpolynomialAt

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: ι : Type u_4
inst✝ : DecidableEq ι
n : ℕ
e : Fin (n + 1) ↪ ι
k : ι
h'k : k ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hk : k ∈ range ⇑e
⊢ (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨k, h'k⟩).succ = e.toEquivRange.symm ⟨k, hk⟩

Tactic:
rcases hk with ⟨j, rfl⟩
  have hj : j ≠ 0 := by
    rintro rfl
    simp at h'k
  simp only [Function.Embedding.toEquivRange_symm_apply_self]
  have : e j = (Equiv.embeddingFinSucc n ι e).1 (Fin.pred j hj) := by simp
  simp_rw [this]
  simp [-Equiv.embeddingFinSucc_fst]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Analytic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ
x : (i : ι) → E i
⊢ ‖f.iteratedFDeriv n x‖ ≤ ↑((Fintype.card ι).descFactorial n) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - n)

Tactic:
exact f.norm_iteratedFDeriv_le' n x

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
b : 𝕜
⊢ (b - b) • dslope f b b = f b - f b

Tactic:
simp [dslope_of_ne, *]

Goal state after:
Module: Mathlib.Analysis.Calculus.DSlope

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
a b : 𝕜
hne : b ≠ a
⊢ (b - a) • dslope f a b = f b - f a

Tactic:
simp [dslope_of_ne, *]

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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
h' : 𝕜'
g₁ : 𝕜' → F
g₁' : F
y : 𝕜'
hg : HasDerivWithinAt g₁ g₁' t' y
hh : HasDerivWithinAt h h' s x
hst : MapsTo h s t'
hy : y = h x
⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x

Tactic:
by
  rw [hy] at hg; exact hg.scomp x hh hst

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

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

Tactic:
· exact (hc.hasDerivWithinAt.div hd.hasDerivWithinAt hx).derivWithin hsx

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: 𝕜 : Type u_3
G : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → G
hf : Differentiable 𝕜 f
hf' : ∀ (x : E), fderiv 𝕜 f x = 0
x y : E
⊢ f x = f y

Tactic:
let : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
  let A : NormedSpace ℝ E := .restrictScalars ℝ 𝕜 E
  exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn
    (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial

Goal state after:
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DifferentiableOn 𝕜 f s
hf' : EqOn (fderiv 𝕜 f) 0 s
t : Set G
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ f ⁻¹' t)

Tactic:
by
  refine Metric.isOpen_iff.mpr fun y ⟨hy, hy'⟩ ↦ ?_
  obtain ⟨r, hr, h⟩ := Metric.isOpen_iff.mp hs y hy
  refine ⟨r, hr, Set.subset_inter h fun x hx ↦ ?_⟩
  have := (convex_ball y r).is_const_of_fderivWithin_eq_zero (hf.mono h) ?_ hx (mem_ball_self hr)
  · simpa [this]
  · intro z hz
    simpa only [fderivWithin_of_isOpen Metric.isOpen_ball hz] using! hf' (h hz)

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
x : E
n : ℕ∞ω
y : E
⊢ ContDiffWithinAt 𝕜 n f (s \ {y}) x ↔ ContDiffWithinAt 𝕜 n f s x

Tactic:
by
  rw [← contDiffWithinAt_insert, insert_sdiff_singleton, contDiffWithinAt_insert]

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

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
f g : 𝕜 → 𝕜
a : 𝕜
s t : Set 𝕜
ha : a ∈ s
hsu : UniqueDiffWithinAt 𝕜 s a
hf : HasDerivWithinAt f 0 t (g a)
hst : MapsTo g s t
hfg : f ∘ g =ᶠ[𝓝[s] a] id
⊢ ¬DifferentiableWithinAt 𝕜 g s a

Tactic:
by
  intro hg
  have := (hf.comp a hg.hasDerivWithinAt hst).congr_of_eventuallyEq_of_mem hfg.symm ha
  simpa using hsu.eq_deriv _ this (hasDerivWithinAt_id _ _)

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

Goal state before:
pp: case refine_1.refine_2
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 ≠ ∞
hf : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
f_an : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
huf' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
hf' : ContDiffWithinAt 𝕜 n f' u x
w : Set E
hw : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] w
hxw : x ∈ w
hwu : insert x s ∩ w ⊆ u
y : E
hy : y ∈ insert x s ∩ w
⊢ insert x s ∩ w ∈ 𝓝[s] y

Tactic:
grw [← subset_insert]

Goal state after:
case refine_1.refine_2
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 ≠ ∞
hf : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
f_an : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
huf' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
hf' : ContDiffWithinAt 𝕜 n f' u x
w : Set E
hw : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] w
hxw : x ∈ w
hwu : insert x s ∩ w ⊆ u
y : E
hy : y ∈ insert x s ∩ w
⊢ s ∩ w ∈ 𝓝[s] y
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
⊢ ContDiffOn 𝕜 (↑n) f s

Tactic:
intro x hx m hm

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
m : ℕ
hm : ↑m ≤ n
⊢ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) f p u
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case inl
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
m : ℕ∞ω
hm : m ≤ ω
h' : m = ∞ → ω = ω
t : Set E
ht : t ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p t
h'p : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) t
u : Set E
huo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
hxu : x ∈ u
hut : insert x s ∩ u ⊆ t
⊢ ContDiffOn 𝕜 ω f (insert x s ∩ u)

Tactic:
intro y

Goal state after:
case inl
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
m : ℕ∞ω
hm : m ≤ ω
h' : m = ∞ → ω = ω
t : Set E
ht : t ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p t
h'p : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) t
u : Set E
huo : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
hxu : x ∈ u
hut : insert x s ∩ u ⊆ t
y : E
⊢ y ∈ insert x s ∩ u → ContDiffWithinAt 𝕜 ω f (insert x s ∩ u) y
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
⊢ iteratedDerivWithin n f s = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDerivWithin 𝕜 n f s

Tactic:
ext x; rfl

Goal state after:
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
⊢ iteratedDeriv n f = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f

Tactic:
ext x; rfl

Goal state after: