mathlib-initiative/mathlib-tactics huggingface dataset:
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s
m : ℕ
hm : ↑m < ↑n
⊢ ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ (p x).shift m) ((p x).shift m.succ).curryLeft s x

Tactic:
x

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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s
m : ℕ
hm : ↑m < ↑n
x : E
⊢ x ∈ s → HasFDerivWithinAt (fun x ↦ (p x).shift m) ((p x).shift m.succ).curryLeft s x
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
Hzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
Hfderiv_zero : ∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x
Htaylor : HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
m : ℕ
hm : ↑m < ↑n.succ
⊢ ∀ x ∈ s, HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x

Tactic:
x

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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
Hzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
Hfderiv_zero : ∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x
Htaylor : HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
m : ℕ
hm : ↑m < ↑n.succ
x : E
⊢ x ∈ s → HasFDerivWithinAt (fun x ↦ p x m) (p x m.succ).curryLeft s x
Module: Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Goal state before:
pp: case fderivWithin.succ
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
Hzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
Hfderiv_zero : ∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x
Htaylor : HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑(m + 1) < ↑n.succ
A : ↑m < ↑n
this :
  HasFDerivWithinAt (fun x ↦ p x m.succ)
    (↑↑(continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘SL ((p x).shift m.succ).curryLeft) s x
⊢ HasFDerivWithinAt (fun x ↦ p x (m + 1)) (p x (m + 1).succ).curryLeft s x

Tactic:
convert! this

Goal state after:
case e'_12
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
Hzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
Hfderiv_zero : ∀ x ∈ s, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft s x
Htaylor : HasFTaylorSeriesUpToOn (↑n) (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) s
x : E
hx : x ∈ s
m : ℕ
hm : ↑(m + 1) < ↑n.succ
A : ↑m < ↑n
this :
  HasFDerivWithinAt (fun x ↦ p x m.succ)
    (↑↑(continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘SL ((p x).shift m.succ).curryLeft) s x
e_8✝ : ContinuousMultilinearMap.instAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'.toAddCommGroup
e_9✝ : ContinuousMultilinearMap.instModule ≍ ContinuousMultilinearMap.normedSpace'.toModule
e_10✝ : ContinuousMultilinearMap.instTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ (p x (m + 1).succ).curryLeft =
    ↑↑(continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘SL ((p x).shift m.succ).curryLeft
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)
⊢ HasSum (fun n ↦ (p.derivSeries n) fun x ↦ y) f'

Tactic:
by
  /- In the completion of the space, the derivative series is summable, and its sum is a derivative
  of the function. Therefore, by uniqueness of derivatives, its sum is the image of `f'` under
  the canonical embedding. As this is an embedding, it means that there was also convergence in
  the original space, to `f'`. -/
  let F' := UniformSpace.Completion F
  let a : F →L[𝕜] F' := UniformSpace.Completion.toComplL
  let b : (E →L[𝕜] F) →ₗᵢ[𝕜] (E →L[𝕜] F') := UniformSpace.Completion.toComplₗᵢ.postcomp
  rw [← b.isEmbedding.hasSum_iff]
  have : HasFPowerSeriesWithinOnBall (a ∘ f) (a.compFormalMultilinearSeries p) s x r :=
    a.comp_hasFPowerSeriesWithinOnBall h
  have Z := (this.fderivWithin hu).hasSum h'y (by simpa [edist_zero_right] using! hy)
  have : fderivWithin 𝕜 (a ∘ f) (insert x s) (x + y) = a ∘L f' := by
    apply HasFDerivWithinAt.fderivWithin _ (hu _ h'y)
    exact a.hasFDerivAt.comp_hasFDerivWithinAt (x + y) hf'
  rw [this] at Z
  convert! Z with n
  ext v
  simp only [FormalMultilinearSeries.derivSeries, sum_apply,
    ContinuousLinearMap.compFormalMultilinearSeries_apply,
    FormalMultilinearSeries.changeOriginSeries,
    ContinuousLinearMap.compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe,
    LinearIsometryEquiv.coe_coe, Function.comp_apply, sum_apply, map_sum]
  rfl

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
c : F
x✝¹ : Set 𝕜
x✝ : 𝕜
⊢ derivWithin (fun x ↦ f x - c) x✝¹ x✝ = derivWithin f x✝¹ x✝

Tactic:
apply derivWithin_sub_const

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
m : ℕ
hm : ¬m = Fintype.card ι - 1
⊢ 1 ≤ Fintype.card ι

Tactic:
by exact Fintype.card_pos

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

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
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
δ : ℝ
δ_pos : δ > 0
δε : Metric.ball 0 δ ⊆ t
n : ℕ
p : E [×n + 1]→L[𝕜] F
ht : ∀ y ∈ t, ‖p fun x ↦ y‖ ≤ c * ‖‖y‖ ^ (n + 1 + 1)‖
hy : 0 < ‖y‖
ε : ℝ
ε_pos : 0 < ε
h₀ : 0 < c * ‖y‖ ^ (n.succ + 1)
k : 𝕜
k_pos : 0 < ‖k‖
k_norm : ‖k‖ < min (δ * ‖y‖⁻¹) (ε * (c * ‖y‖ ^ (n.succ + 1))⁻¹)
h₁ : ‖k • y‖ < δ
h₂ : ‖p fun x ↦ k • y‖ ≤ ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))
h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε
⊢ ‖k⁻¹ ^ n.succ‖ * ‖p fun x ↦ k • y‖ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))))

Tactic:
by gcongr

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
⊢ HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f')

Tactic:
simp only [HasDerivWithinAt, nhdsWithin, sdiff_eq, ← inf_assoc, inf_principal.symm]

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
s : Set 𝕜
⊢ HasDerivAtFilter f f' ((𝓝 x ⊓ 𝓟 s) ×ˢ pure x) ↔ Tendsto (slope f x) (𝓝 x ⊓ 𝓟 s ⊓ 𝓟 {x}ᶜ) (𝓝 f')
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
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
h : p ≠ 0
⊢ (swap dslope z₀)^[p.order] f z₀ ≠ 0

Tactic:
rw [← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1]
  simpa [coeff_eq_zero] using apply_order_ne_zero h

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
L : Filter (𝕜 × 𝕜)
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
h' h₂' : 𝕜'
L' : Filter (𝕜' × 𝕜')
hh₂ : HasDerivAtFilter h₂ h₂' L'
hh : HasDerivAtFilter h h' L
hL : Tendsto (Prod.map h h) L L'
⊢ HasDerivAtFilter (h₂ ∘ h) (h₂' * h') L

Tactic:
rw [mul_comm]
  exact hh₂.scomp hh hL

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

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 →ᴬ[𝕜] F
L : Filter (E × E)
⊢ ∀ (x : E × E), 0 x = f x.1 - f x.2 - f.contLinear (x.1 - x.2)

Tactic:
simp [(vsub_eq_sub _ _).symm.trans (f.contLinear_map_vsub _ _).symm]

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

Goal state before:
pp: case cons
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : ∀ i ∈ Finset.cons a s ha, f i x ≠ 0
hd : ∀ i ∈ Finset.cons a s ha, DifferentiableAt 𝕜 (f i) x
⊢ logDeriv (fun x ↦ ∏ i ∈ Finset.cons a s ha, f i x) x = ∑ i ∈ Finset.cons a s ha, logDeriv (f i) x

Tactic:
rw [Finset.forall_mem_cons] at hf hd

Goal state after:
case cons
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
f : ι → 𝕜 → 𝕜'
x : 𝕜
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x ↦ ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
hf : f a x ≠ 0 ∧ ∀ x_1 ∈ s, f x_1 x ≠ 0
hd : DifferentiableAt 𝕜 (f a) x ∧ ∀ x_1 ∈ s, DifferentiableAt 𝕜 (f x_1) x
⊢ logDeriv (fun x ↦ ∏ i ∈ Finset.cons a s ha, f i x) x = ∑ i ∈ Finset.cons a s ha, logDeriv (f i) x
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
s : Set E
f : E → F
n : ℕ∞ω
inst✝ : Subsingleton F
⊢ ContDiffOn 𝕜 n f s

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

Goal state after:
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 g : E → G
hf : Differentiable 𝕜 f
hg : Differentiable 𝕜 g
hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x
x : E
hfgx : f x = g x
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ f = g

Tactic:
suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x

Goal state after:
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 g : E → G
hf : Differentiable 𝕜 f
hg : Differentiable 𝕜 g
hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x
x : E
hfgx : f x = g x
this : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ EqOn f g univ
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 ⁻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:
x

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
⊢ 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.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 ∈ s, HasFDerivWithinAt (fun x ↦ (p x m).prod (q x m)) ((p x m.succ).prod (q x m.succ)).curryLeft s x

Tactic:
x

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
⊢ 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.Deriv.Inverse

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
f' : F
x : 𝕜
c : F
h : HasDerivAt f f' x
hf' : f' ≠ 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ c

Tactic:
by
  simpa only [compl_eq_univ_sdiff] using (hasDerivWithinAt_univ.2 h).eventually_ne hf'

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case left
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e e' : OpenPartialHomeomorph X Y
h : EqOn (↑e) (↑e') (e.source ∩ e'.source)
⊢ e.source ∩ e'.source = e'.source ∩ e.source

Tactic:
exact Set.inter_comm _ _

Goal state after:
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 ∈ 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:
x

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
⊢ 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
⊢ ContDiffOn 𝕜 (n + 1) f s

Tactic:
x

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
⊢ x ∈ s → ContDiffWithinAt 𝕜 (n + 1) f s 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 ∈ s, HasFDerivWithinAt (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (ftaylorSeriesWithin 𝕜 f s x m.succ).curryLeft s x

Tactic:
x

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
⊢ 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
hx : x ∈ s
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : o ∩ insert x s ⊆ u
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)

Tactic:
rw [insert_eq_of_mem hx] at ho

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
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑m) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : o ∩ s ⊆ u
⊢ ∃ t,
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t ∧
      x ∈ t ∧ ContinuousOn (fun x ↦ ftaylorSeriesWithin 𝕜 f s x m) (s ∩ t)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

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
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) = {0}
this : c.length - 1 + 1 = c.length
⊢ Fin (c.length - 1) → ℕ

Tactic:
exact fun i ↦ c.partSize (Fin.cast this (succ i))

Goal state after:
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
hf : DifferentiableOn 𝕜 f s
h' : ∞ = ω → AnalyticOn 𝕜 f s
h : ContDiffOn 𝕜 ∞ (fun y ↦ fderivWithin 𝕜 f s y) s
m : ℕ
⊢ ContDiffOn 𝕜 (↑m) f s

Tactic:
x

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
hf : DifferentiableOn 𝕜 f s
h' : ∞ = ω → AnalyticOn 𝕜 f s
h : ContDiffOn 𝕜 ∞ (fun y ↦ fderivWithin 𝕜 f s y) s
m : ℕ
x : E
⊢ x ∈ s → ContDiffWithinAt 𝕜 (↑m) 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
n : ℕ∞ω
hs : UniqueDiffOn 𝕜 s
H : ContDiffOn 𝕜 (n + 1) f s
x : E
hx : x ∈ s
m : ℕ
hm : ↑m ≤ n
⊢ ↑m + 1 ≤ n + 1

Tactic:
by gcongr

Goal state after:
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
f : E → F
x : E
n : ℕ
u : Set E
H : u ∈ 𝓝 x
f' : E → E →L[𝕜] F
h_fderiv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
h_cont_diff : ContDiffWithinAt 𝕜 (↑n) f' u x
t : Set E
htu : t ⊆ u
ht : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
hxt : x ∈ t
y : E
hyt : y ∈ t
⊢ u ∈ 𝓝 y

Tactic:
exact mem_nhds_iff.mpr ⟨t, htu, ht, hyt⟩

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
f : E → F
n : ℕ∞ω
⊢ ContDiffOn 𝕜 ω f univ ↔ ContDiff 𝕜 ω f

Tactic:
constructor
    · intro H
      use ftaylorSeriesWithin 𝕜 f univ
      rw [← hasFTaylorSeriesUpToOn_univ_iff]
      refine ⟨H.ftaylorSeriesWithin uniqueDiffOn_univ, fun i ↦ ?_⟩
      rw [← analyticOn_univ]
      exact H.analyticOn.iteratedFDerivWithin uniqueDiffOn_univ _
    · rintro ⟨p, hp, h'p⟩ x _
      exact ⟨univ, Filter.univ_sets _, p, (hp.hasFTaylorSeriesUpToOn univ).of_le le_top,
        fun i ↦ (h'p i).analyticOn⟩

Goal state after: