ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.EqOn.iteratedFDerivWithin
protected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f₁ f s) (n : ℕ) :
    EqOn (iteratedFDerivWithin 𝕜 n f₁ s) (iteratedFDerivWithin 𝕜 n f s) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f f₁ : E → F} (hs : Set.EqOn f₁ f s) (n : ℕ)
Goal: Set.EqOn (iteratedFDerivWithin 𝕜 n f₁ s) (iteratedFDerivWithin 𝕜 n f s) s
-- iteratedFDerivWithin_eventually_congr_set'
theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s t : Set E} {f : E → F} {x : E} (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
-- iteratedFDerivWithin_eventually_congr_set
theorem iteratedFDerivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s t : Set E} {f : E → F} {x : E} (h : s =ᶠ[𝓝 x] t) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t
-- iteratedFDerivWithin_congr_set
theorem iteratedFDerivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s t : Set E} {f : E → F} {x : E} (h : s =ᶠ[𝓝 x] t) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f t x
-- iteratedFDerivWithin_inter'
theorem iteratedFDerivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s u : Set E} {f : E → F} {x : E} {n : ℕ} (hu : u ∈ 𝓝[s] x)
Goal: iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x
-- iteratedFDerivWithin_inter
theorem iteratedFDerivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s u : Set E} {f : E → F} {x : E} {n : ℕ} (hu : u ∈ 𝓝 x)
Goal: iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x
-- iteratedFDerivWithin_inter_open
theorem iteratedFDerivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s u : Set E} {f : E → F} {x : E} {n : ℕ} (hu : IsOpen u) (hx : x ∈ u)
Goal: iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x
-- contDiffOn_zero
@[simp]
theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F}
Goal: ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s
-- contDiffWithinAt_zero
theorem contDiffWithinAt_zero (hx : x ∈ s) :
    ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, ContinuousOn f (s ∩ u) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} (hx : x ∈ s)
Goal: ContDiffWithinAt 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, ContinuousOn f (s ∩ u)
-- HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
theorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
    (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
    (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : ↑m ≤ n) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: p x m = iteratedFDerivWithin 𝕜 m f s x
-- ContDiffOn.ftaylorSeriesWithin
protected theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) :
    HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
Goal: HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s
-- contDiffOn_of_continuousOn_differentiableOn
theorem contDiffOn_of_continuousOn_differentiableOn
    (Hcont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s)
    (Hdiff : ∀ m : ℕ, (m : ℕ∞) < n →
      DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s) :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) (Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s)
Goal: ContDiffOn 𝕜 n f s
-- contDiffOn_of_differentiableOn
theorem contDiffOn_of_differentiableOn
    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s) :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (h : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s)
Goal: ContDiffOn 𝕜 n f s
-- ContDiffOn.continuousOn_iteratedFDerivWithin
theorem ContDiffOn.continuousOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
    (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedFDerivWithin 𝕜 m f s) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s) (hmn : ↑m ≤ n) (hs : UniqueDiffOn 𝕜 s)
Goal: ContinuousOn (iteratedFDerivWithin 𝕜 m f s) s
-- ContDiffOn.differentiableOn_iteratedFDerivWithin
theorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : ℕ} (h : ContDiffOn 𝕜 n f s)
    (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 s) :
    DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s) (hmn : ↑m < n) (hs : UniqueDiffOn 𝕜 s)
Goal: DifferentiableOn 𝕜 (iteratedFDerivWithin 𝕜 m f s) s
-- ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin
theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
    (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
    DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} {m : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) (hmn : ↑m < n) (hs : UniqueDiffOn 𝕜 (insert x s))
Goal: DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x
-- contDiffOn_iff_continuousOn_differentiableOn
theorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 n f s ↔
      (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) ∧
        ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 n f s ↔
  (∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) ∧
    ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s
-- contDiffOn_succ_of_fderivWithin
theorem contDiffOn_succ_of_fderivWithin {n : ℕ} (hf : DifferentiableOn 𝕜 f s)
    (h : ContDiffOn 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) : ContDiffOn 𝕜 (n + 1 : ℕ) f s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ} (hf : DifferentiableOn 𝕜 f s) (h : ContDiffOn 𝕜 (↑n) (fun y => fderivWithin 𝕜 f s y) s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s
-- contDiffOn_succ_iff_fderivWithin
theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
      DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderivWithin 𝕜 f s y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (↑n) (fun y => fderivWithin 𝕜 f s y) s
-- contDiffOn_succ_iff_hasFDerivWithin
theorem contDiffOn_succ_iff_hasFDerivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
      ∃ f' : E → E →L[𝕜] F, ContDiffOn 𝕜 n f' s ∧ ∀ x, x ∈ s → HasFDerivWithinAt f (f' x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s ↔ ∃ f', ContDiffOn 𝕜 (↑n) f' s ∧ ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
-- contDiffOn_succ_iff_fderiv_of_isOpen
theorem contDiffOn_succ_iff_fderiv_of_isOpen {n : ℕ} (hs : IsOpen s) :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
      DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderiv 𝕜 f y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ} (hs : IsOpen s)
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (↑n) (fun y => fderiv 𝕜 f y) s
-- contDiffOn_top_iff_fderivWithin
theorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 ∞ f s ↔
      DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderivWithin 𝕜 f s y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 ⊤ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderivWithin 𝕜 f s y) s
-- contDiffOn_top_iff_fderiv_of_isOpen
theorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :
    ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fun y => fderiv 𝕜 f y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} (hs : IsOpen s)
Goal: ContDiffOn 𝕜 ⊤ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderiv 𝕜 f y) s
-- ContDiffOn.fderivWithin
protected theorem ContDiffOn.fderivWithin (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
    (hmn : m + 1 ≤ n) : ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {m n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hmn : m + 1 ≤ n)
Goal: ContDiffOn 𝕜 m (fun y => fderivWithin 𝕜 f s y) s
-- ContDiffOn.fderiv_of_isOpen
theorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
    ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {m n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n)
Goal: ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s
-- ContDiffOn.continuousOn_fderivWithin
theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
    (hn : 1 ≤ n) : ContinuousOn (fun x => fderivWithin 𝕜 f s x) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hn : 1 ≤ n)
Goal: ContinuousOn (fun x => fderivWithin 𝕜 f s x) s
-- ContDiffOn.continuousOn_fderiv_of_isOpen
theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s)
    (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hn : 1 ≤ n)
Goal: ContinuousOn (fun x => fderiv 𝕜 f x) s
-- HasFTaylorSeriesUpTo.zero_eq
structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
  Prop where
  zero_eq : ∀ x, (p x 0).uncurry0 = f x
  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} (self : HasFTaylorSeriesUpTo n f p) (x : E)
Goal: (p x 0).uncurry0 = f x
-- HasFTaylorSeriesUpTo.fderiv
structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
  Prop where
  zero_eq : ∀ x, (p x 0).uncurry0 = f x
  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} (self : HasFTaylorSeriesUpTo n f p) (m : ℕ)
Goal: ↑m < n → ∀ (x : E), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
-- HasFTaylorSeriesUpTo.cont
structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
  Prop where
  zero_eq : ∀ x, (p x 0).uncurry0 = f x
  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} (self : HasFTaylorSeriesUpTo n f p) (m : ℕ)
Goal: ↑m ≤ n → Continuous fun x => p x m
-- HasFTaylorSeriesUpTo.zero_eq'
theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :
    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) (x : E)
Goal: p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x)
-- hasFTaylorSeriesUpToOn_univ_iff
theorem hasFTaylorSeriesUpToOn_univ_iff :
    HasFTaylorSeriesUpToOn n f p univ ↔ HasFTaylorSeriesUpTo n f p 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpToOn n f p Set.univ ↔ HasFTaylorSeriesUpTo n f p
-- HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn
theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :
    HasFTaylorSeriesUpToOn n f p s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) (s : Set E)
Goal: HasFTaylorSeriesUpToOn n f p s
-- HasFTaylorSeriesUpTo.ofLe
theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
    HasFTaylorSeriesUpTo m f p 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {m n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n)
Goal: HasFTaylorSeriesUpTo m f p
-- HasFTaylorSeriesUpTo.continuous
theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p)
Goal: Continuous f
-- hasFTaylorSeriesUpTo_zero_iff
theorem hasFTaylorSeriesUpTo_zero_iff :
    HasFTaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ x, (p x 0).uncurry0 = f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ (x : E), (p x 0).uncurry0 = f x
-- hasFTaylorSeriesUpTo_top_iff
theorem hasFTaylorSeriesUpTo_top_iff :
    HasFTaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFTaylorSeriesUpTo n f p 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpTo ⊤ f p ↔ ∀ (n : ℕ), HasFTaylorSeriesUpTo (↑n) f p
-- hasFTaylorSeriesUpTo_top_iff'
theorem hasFTaylorSeriesUpTo_top_iff' :
    HasFTaylorSeriesUpTo ∞ f p ↔
      (∀ x, (p x 0).uncurry0 = f x) ∧
        ∀ (m : ℕ) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpTo ⊤ f p ↔
  (∀ (x : E), (p x 0).uncurry0 = f x) ∧ ∀ (m : ℕ) (x : E), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
-- HasFTaylorSeriesUpTo.hasFDerivAt
theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E)
Goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) x
-- HasFTaylorSeriesUpTo.differentiable
theorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
    Differentiable 𝕜 f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n)
Goal: Differentiable 𝕜 f
-- hasFTaylorSeriesUpTo_succ_iff_right
theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
    HasFTaylorSeriesUpTo (n + 1 : ℕ) f p ↔
      (∀ x, (p x 0).uncurry0 = f x) ∧
        (∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
          HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
            (p x).shift 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} {n : ℕ}
Goal: HasFTaylorSeriesUpTo (↑(n + 1)) f p ↔
  (∀ (x : E), (p x 0).uncurry0 = f x) ∧
    (∀ (x : E), HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
      HasFTaylorSeriesUpTo (↑n) (fun x => (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) fun x => (p x).shift
-- contDiffWithinAt_univ
theorem contDiffWithinAt_univ : ContDiffWithinAt 𝕜 n f univ x ↔ ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {n : ℕ∞}
Goal: ContDiffWithinAt 𝕜 n f Set.univ x ↔ ContDiffAt 𝕜 n f x
-- contDiffAt_top
theorem contDiffAt_top : ContDiffAt 𝕜 ∞ f x ↔ ∀ n : ℕ, ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E}
Goal: ContDiffAt 𝕜 ⊤ f x ↔ ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) f x
-- ContDiffAt.contDiffWithinAt
theorem ContDiffAt.contDiffWithinAt (h : ContDiffAt 𝕜 n f x) : ContDiffWithinAt 𝕜 n f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffAt 𝕜 n f x)
Goal: ContDiffWithinAt 𝕜 n f s x
-- ContDiffWithinAt.contDiffAt
theorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt 𝕜 n f s x) (hx : s ∈ 𝓝 x) :
    ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) (hx : s ∈ 𝓝 x)
Goal: ContDiffAt 𝕜 n f x
-- ContDiffOn.contDiffAt
theorem ContDiffOn.contDiffAt (h : ContDiffOn 𝕜 n f s) (hx : s ∈ 𝓝 x) :
    ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (hx : s ∈ 𝓝 x)
Goal: ContDiffAt 𝕜 n f x
-- ContDiffAt.congr_of_eventuallyEq
theorem ContDiffAt.congr_of_eventuallyEq (h : ContDiffAt 𝕜 n f x) (hg : f₁ =ᶠ[𝓝 x] f) :
    ContDiffAt 𝕜 n f₁ x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {n : ℕ∞} (h : ContDiffAt 𝕜 n f x) (hg : f₁ =ᶠ[𝓝 x] f)
Goal: ContDiffAt 𝕜 n f₁ x
-- ContDiffAt.of_le
theorem ContDiffAt.of_le (h : ContDiffAt 𝕜 n f x) (hmn : m ≤ n) : ContDiffAt 𝕜 m f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {m n : ℕ∞} (h : ContDiffAt 𝕜 n f x) (hmn : m ≤ n)
Goal: ContDiffAt 𝕜 m f x