ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffAt.continuousAt
theorem ContDiffAt.continuousAt (h : ContDiffAt 𝕜 n f x) : ContinuousAt 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∞} (h : ContDiffAt 𝕜 n f x)
Goal: ContinuousAt f x
-- ContDiffAt.differentiableAt
theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
    DifferentiableAt 𝕜 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∞} (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n)
Goal: DifferentiableAt 𝕜 f x
-- ContDiffAt.contDiffOn
nonrec lemma ContDiffAt.contDiffOn {m : ℕ} (h : ContDiffAt 𝕜 n f x) (hm : m ≤ n) :
    ∃ u ∈ 𝓝 x, ContDiffOn 𝕜 m f u 
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∞} {m : ℕ} (h : ContDiffAt 𝕜 n f x) (hm : ↑m ≤ n)
Goal: ∃ u ∈ 𝓝 x, ContDiffOn 𝕜 (↑m) f u
-- contDiffAt_succ_iff_hasFDerivAt
theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
    ContDiffAt 𝕜 (n + 1 : ℕ) f x ↔
      ∃ f' : E → E →L[𝕜] F, (∃ u ∈ 𝓝 x, ∀ x ∈ u, HasFDerivAt f (f' x) 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: ContDiffAt 𝕜 (↑(n + 1)) f x ↔ ∃ f', (∃ u ∈ 𝓝 x, ∀ x ∈ u, HasFDerivAt f (f' x) x) ∧ ContDiffAt 𝕜 (↑n) f' x
-- ContDiffAt.eventually
protected theorem ContDiffAt.eventually {n : ℕ} (h : ContDiffAt 𝕜 n f x) :
    ∀ᶠ y in 𝓝 x, ContDiffAt 𝕜 n f y 
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} (h : ContDiffAt 𝕜 (↑n) f x)
Goal: ∀ᶠ (y : E) in 𝓝 x, ContDiffAt 𝕜 (↑n) f y
-- HasFTaylorSeriesUpTo.contDiff
theorem HasFTaylorSeriesUpTo.contDiff {f' : E → FormalMultilinearSeries 𝕜 E F}
    (hf : HasFTaylorSeriesUpTo n f f') : ContDiff 𝕜 n 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∞} {f' : E → FormalMultilinearSeries 𝕜 E F} (hf : HasFTaylorSeriesUpTo n f f')
Goal: ContDiff 𝕜 n f
-- contDiffOn_univ
theorem contDiffOn_univ : ContDiffOn 𝕜 n f univ ↔ ContDiff 𝕜 n 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∞}
Goal: ContDiffOn 𝕜 n f Set.univ ↔ ContDiff 𝕜 n f
-- contDiff_iff_contDiffAt
theorem contDiff_iff_contDiffAt : ContDiff 𝕜 n f ↔ ∀ 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} {n : ℕ∞}
Goal: ContDiff 𝕜 n f ↔ ∀ (x : E), ContDiffAt 𝕜 n f x
-- ContDiff.contDiffAt
theorem ContDiff.contDiffAt (h : ContDiff 𝕜 n f) : 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∞} (h : ContDiff 𝕜 n f)
Goal: ContDiffAt 𝕜 n f x
-- ContDiff.contDiffWithinAt
theorem ContDiff.contDiffWithinAt (h : ContDiff 𝕜 n f) : 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 : ContDiff 𝕜 n f)
Goal: ContDiffWithinAt 𝕜 n f s x
-- contDiff_top
theorem contDiff_top : ContDiff 𝕜 ∞ f ↔ ∀ n : ℕ, ContDiff 𝕜 n 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}
Goal: ContDiff 𝕜 ⊤ f ↔ ∀ (n : ℕ), ContDiff 𝕜 (↑n) f
-- contDiff_all_iff_nat
theorem contDiff_all_iff_nat : (∀ n, ContDiff 𝕜 n f) ↔ ∀ n : ℕ, ContDiff 𝕜 n 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}
Goal: (∀ (n : ℕ∞), ContDiff 𝕜 n f) ↔ ∀ (n : ℕ), ContDiff 𝕜 (↑n) f
-- ContDiff.contDiffOn
theorem ContDiff.contDiffOn (h : ContDiff 𝕜 n f) : 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 : ContDiff 𝕜 n f)
Goal: ContDiffOn 𝕜 n f s
-- contDiff_zero
@[simp]
theorem contDiff_zero : ContDiff 𝕜 0 f ↔ 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}
Goal: ContDiff 𝕜 0 f ↔ Continuous f
-- contDiffAt_zero
theorem contDiffAt_zero : ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u 
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 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u
-- contDiffAt_one_iff
theorem contDiffAt_one_iff :
    ContDiffAt 𝕜 1 f x ↔
      ∃ f' : E → E →L[𝕜] F, ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) 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 1 f x ↔ ∃ f', ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x
-- ContDiff.of_le
theorem ContDiff.of_le (h : ContDiff 𝕜 n f) (hmn : m ≤ n) : ContDiff 𝕜 m 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} {m n : ℕ∞} (h : ContDiff 𝕜 n f) (hmn : m ≤ n)
Goal: ContDiff 𝕜 m f
-- ContDiff.of_succ
theorem ContDiff.of_succ {n : ℕ} (h : ContDiff 𝕜 (n + 1) f) : ContDiff 𝕜 n 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} (h : ContDiff 𝕜 (↑n + 1) f)
Goal: ContDiff 𝕜 (↑n) f
-- ContDiff.one_of_succ
theorem ContDiff.one_of_succ {n : ℕ} (h : ContDiff 𝕜 (n + 1) f) : ContDiff 𝕜 1 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} (h : ContDiff 𝕜 (↑n + 1) f)
Goal: ContDiff 𝕜 1 f
-- ContDiff.continuous
theorem ContDiff.continuous (h : ContDiff 𝕜 n f) : 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∞} (h : ContDiff 𝕜 n f)
Goal: Continuous f
-- ContDiff.differentiable
theorem ContDiff.differentiable (h : ContDiff 𝕜 n f) (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∞} (h : ContDiff 𝕜 n f) (hn : 1 ≤ n)
Goal: Differentiable 𝕜 f
-- contDiff_iff_forall_nat_le
theorem contDiff_iff_forall_nat_le : ContDiff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤ n → ContDiff 𝕜 m 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∞}
Goal: ContDiff 𝕜 n f ↔ ∀ (m : ℕ), ↑m ≤ n → ContDiff 𝕜 (↑m) f
-- contDiff_succ_iff_hasFDerivAt
theorem contDiff_succ_iff_hasFDerivAt {n : ℕ} :
    ContDiff 𝕜 (n + 1 : ℕ) f ↔
      ∃ f' : E → E →L[𝕜] F, ContDiff 𝕜 n f' ∧ ∀ x, HasFDerivAt f (f' x) 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}
Goal: ContDiff 𝕜 (↑(n + 1)) f ↔ ∃ f', ContDiff 𝕜 (↑n) f' ∧ ∀ (x : E), HasFDerivAt f (f' x) x
-- contDiff_one_iff_hasFDerivAt
theorem contDiff_one_iff_hasFDerivAt : ContDiff 𝕜 1 f ↔
    ∃ f' : E → E →L[𝕜] F, Continuous f' ∧ ∀ x, HasFDerivAt f (f' x) 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}
Goal: ContDiff 𝕜 1 f ↔ ∃ f', Continuous f' ∧ ∀ (x : E), HasFDerivAt f (f' x) x
-- iteratedFDeriv_zero_apply
@[simp]
theorem iteratedFDeriv_zero_apply (m : Fin 0 → E) :
    (iteratedFDeriv 𝕜 0 f x : (Fin 0 → E) → F) 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 : Fin 0 → E)
Goal: (iteratedFDeriv 𝕜 0 f x) m = f x
-- iteratedFDeriv_zero_eq_comp
theorem iteratedFDeriv_zero_eq_comp :
    iteratedFDeriv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ 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}
Goal: iteratedFDeriv 𝕜 0 f = ⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f
-- norm_iteratedFDeriv_zero
@[simp]
theorem norm_iteratedFDeriv_zero : ‖iteratedFDeriv 𝕜 0 f x‖ = ‖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: ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖
-- iteratedFDerivWithin_zero_eq
theorem iteratedFDerivWithin_zero_eq : iteratedFDerivWithin 𝕜 0 f s = iteratedFDeriv 𝕜 0 f 
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: iteratedFDerivWithin 𝕜 0 f s = iteratedFDeriv 𝕜 0 f
-- iteratedFDeriv_succ_apply_left
theorem iteratedFDeriv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
      (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) 
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} (m : Fin (n + 1) → E)
Goal: (iteratedFDeriv 𝕜 (n + 1) f x) m = ((fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x) (m 0)) (Fin.tail m)
-- iteratedFDeriv_succ_eq_comp_left
theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
    iteratedFDeriv 𝕜 (n + 1) f =
      continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F ∘
        fderiv 𝕜 (iteratedFDeriv 𝕜 n 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}
Goal: iteratedFDeriv 𝕜 (n + 1) f = ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)
-- fderiv_iteratedFDeriv
theorem fderiv_iteratedFDeriv {n : ℕ} :
    fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
        iteratedFDeriv 𝕜 (n + 1) 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}
Goal: fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
  ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F).symm ∘ iteratedFDeriv 𝕜 (n + 1) f
-- tsupport_iteratedFDeriv_subset
theorem tsupport_iteratedFDeriv_subset (n : ℕ) : tsupport (iteratedFDeriv 𝕜 n f) ⊆ tsupport 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)
Goal: tsupport (iteratedFDeriv 𝕜 n f) ⊆ tsupport f
-- support_iteratedFDeriv_subset
theorem support_iteratedFDeriv_subset (n : ℕ) : support (iteratedFDeriv 𝕜 n f) ⊆ tsupport 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)
Goal: Function.support (iteratedFDeriv 𝕜 n f) ⊆ tsupport f
-- HasCompactSupport.iteratedFDeriv
theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
    HasCompactSupport (iteratedFDeriv 𝕜 n 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} (hf : HasCompactSupport f) (n : ℕ)
Goal: HasCompactSupport (iteratedFDeriv 𝕜 n f)
-- norm_fderiv_iteratedFDeriv
theorem norm_fderiv_iteratedFDeriv {n : ℕ} :
    ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) 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: ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖
-- iteratedFDerivWithin_univ
theorem iteratedFDerivWithin_univ {n : ℕ} :
    iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n 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}
Goal: iteratedFDerivWithin 𝕜 n f Set.univ = iteratedFDeriv 𝕜 n f
-- HasFTaylorSeriesUpTo.eq_iteratedFDeriv
theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv
    (h : HasFTaylorSeriesUpTo n f p) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (x : E) :
    p x m = iteratedFDeriv 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} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpTo n f p) {m : ℕ} (hmn : ↑m ≤ n) (x : E)
Goal: p x m = iteratedFDeriv 𝕜 m f x
-- iteratedFDerivWithin_of_isOpen
theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
    EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 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) (hs : IsOpen s)
Goal: Set.EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s
-- ftaylorSeriesWithin_univ
theorem ftaylorSeriesWithin_univ : ftaylorSeriesWithin 𝕜 f univ = ftaylorSeries 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}
Goal: ftaylorSeriesWithin 𝕜 f Set.univ = ftaylorSeries 𝕜 f
-- iteratedFDeriv_succ_apply_right
theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
      iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) 
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} (m : Fin (n + 1) → E)
Goal: (iteratedFDeriv 𝕜 (n + 1) f x) m = ((iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x) (Fin.init m)) (m (Fin.last n))
-- iteratedFDeriv_succ_eq_comp_right
theorem iteratedFDeriv_succ_eq_comp_right {n : ℕ} :
    iteratedFDeriv 𝕜 (n + 1) f x =
      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘ iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y)
        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: iteratedFDeriv 𝕜 (n + 1) f x =
  (⇑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y) x
-- norm_iteratedFDeriv_fderiv
theorem norm_iteratedFDeriv_fderiv {n : ℕ} :
    ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) 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: ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖
-- iteratedFDeriv_one_apply
@[simp]
theorem iteratedFDeriv_one_apply (m : Fin 1 → E) :
    iteratedFDeriv 𝕜 1 f x m = fderiv 𝕜 f x (m 0) 
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 : Fin 1 → E)
Goal: (iteratedFDeriv 𝕜 1 f x) m = (fderiv 𝕜 f x) (m 0)
-- iteratedFDeriv_two_apply
lemma iteratedFDeriv_two_apply (f : E → F) (z : E) (m : Fin 2 → E) :
    iteratedFDeriv 𝕜 2 f z m = fderiv 𝕜 (fderiv 𝕜 f) z (m 0) (m 1) 
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) (z : E) (m : Fin 2 → E)
Goal: (iteratedFDeriv 𝕜 2 f z) m = ((fderiv 𝕜 (fderiv 𝕜 f) z) (m 0)) (m 1)
-- contDiff_iff_ftaylorSeries
theorem contDiff_iff_ftaylorSeries :
    ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 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∞}
Goal: ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 𝕜 f)
-- contDiff_iff_continuous_differentiable
theorem contDiff_iff_continuous_differentiable :
    ContDiff 𝕜 n f ↔
      (∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧
        ∀ m : ℕ, (m : ℕ∞) < n → Differentiable 𝕜 fun x => iteratedFDeriv 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} {n : ℕ∞}
Goal: ContDiff 𝕜 n f ↔
  (∀ (m : ℕ), ↑m ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧
    ∀ (m : ℕ), ↑m < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x
-- ContDiff.continuous_iteratedFDeriv
theorem ContDiff.continuous_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) ≤ n) (hf : ContDiff 𝕜 n f) :
    Continuous fun x => iteratedFDeriv 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} {n : ℕ∞} {m : ℕ} (hm : ↑m ≤ n) (hf : ContDiff 𝕜 n f)
Goal: Continuous fun x => iteratedFDeriv 𝕜 m f x
-- ContDiff.differentiable_iteratedFDeriv
theorem ContDiff.differentiable_iteratedFDeriv {m : ℕ} (hm : (m : ℕ∞) < n) (hf : ContDiff 𝕜 n f) :
    Differentiable 𝕜 fun x => iteratedFDeriv 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} {n : ℕ∞} {m : ℕ} (hm : ↑m < n) (hf : ContDiff 𝕜 n f)
Goal: Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x
-- contDiff_of_differentiable_iteratedFDeriv
theorem contDiff_of_differentiable_iteratedFDeriv
    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → Differentiable 𝕜 (iteratedFDeriv 𝕜 m f)) : ContDiff 𝕜 n 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∞} (h : ∀ (m : ℕ), ↑m ≤ n → Differentiable 𝕜 (iteratedFDeriv 𝕜 m f))
Goal: ContDiff 𝕜 n f
-- contDiff_succ_iff_fderiv
theorem contDiff_succ_iff_fderiv {n : ℕ} :
    ContDiff 𝕜 (n + 1 : ℕ) f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 n fun y => fderiv 𝕜 f y 
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}
Goal: ContDiff 𝕜 (↑(n + 1)) f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ↑n fun y => fderiv 𝕜 f y
-- contDiff_one_iff_fderiv
theorem contDiff_one_iff_fderiv : ContDiff 𝕜 1 f ↔ Differentiable 𝕜 f ∧ Continuous (fderiv 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}
Goal: ContDiff 𝕜 1 f ↔ Differentiable 𝕜 f ∧ Continuous (fderiv 𝕜 f)
-- contDiff_top_iff_fderiv
theorem contDiff_top_iff_fderiv :
    ContDiff 𝕜 ∞ f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ∞ fun y => fderiv 𝕜 f y 
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}
Goal: ContDiff 𝕜 ⊤ f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ⊤ fun y => fderiv 𝕜 f y
-- ContDiff.continuous_fderiv
theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
    Continuous fun x => fderiv 𝕜 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∞} (h : ContDiff 𝕜 n f) (hn : 1 ≤ n)
Goal: Continuous fun x => fderiv 𝕜 f x
-- ContDiff.continuous_fderiv_apply
theorem ContDiff.continuous_fderiv_apply (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
    Continuous fun p : E × E => (fderiv 𝕜 f p.1 : E → F) p.2 
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∞} (h : ContDiff 𝕜 n f) (hn : 1 ≤ n)
Goal: Continuous fun p => (fderiv 𝕜 f p.1) p.2
-- Ordinal.isOpen_singleton_iff
theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) ↔ ¬IsLimit a 
Arguments: {a : Ordinal.{u}}
Goal: IsOpen {a} ↔ ¬a.IsLimit
-- Ordinal.nhds_right'
theorem nhds_right' (a : Ordinal) : 𝓝[>] a = ⊥ 
Arguments: (a : Ordinal.{u_1})
Goal: 𝓝[>] a = ⊥
-- Ordinal.nhds_left'_eq_nhds_ne
theorem nhds_left'_eq_nhds_ne (a : Ordinal) : 𝓝[<] a = 𝓝[≠] a 
Arguments: (a : Ordinal.{u_1})
Goal: 𝓝[<] a = 𝓝[≠] a
-- Ordinal.nhds_left_eq_nhds
theorem nhds_left_eq_nhds (a : Ordinal) : 𝓝[≤] a = 𝓝 a 
Arguments: (a : Ordinal.{u_1})
Goal: 𝓝[≤] a = 𝓝 a
-- Ordinal.nhdsBasis_Ioc
theorem nhdsBasis_Ioc (h : a ≠ 0) : (𝓝 a).HasBasis (· < a) (Set.Ioc · a) 
Arguments: {a : Ordinal.{u}} (h : a ≠ 0)
Goal: (𝓝 a).HasBasis (fun x => x < a) fun x => Set.Ioc x a
-- Ordinal.nhds_eq_pure
theorem nhds_eq_pure : 𝓝 a = pure a ↔ ¬IsLimit a 
Arguments: {a : Ordinal.{u}}
Goal: 𝓝 a = pure a ↔ ¬a.IsLimit
-- Ordinal.isOpen_iff
theorem isOpen_iff : IsOpen s ↔ ∀ o ∈ s, IsLimit o → ∃ a < o, Set.Ioo a o ⊆ s 
Arguments: {s : Set Ordinal.{u}}
Goal: IsOpen s ↔ ∀ o ∈ s, o.IsLimit → ∃ a < o, Set.Ioo a o ⊆ s
-- Ordinal.mem_closure_tfae
theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
    TFAE [a ∈ closure s,
      a ∈ closure (s ∩ Iic a),
      (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
      ∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
      ∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
        (∀ x hx, f x hx ∈ s) ∧ bsup.{u, u} o f = a,
      ∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ sup.{u, u} f = a] 
Arguments: (a : Ordinal.{u}) (s : Set Ordinal.{u})
Goal: [a ∈ closure s, a ∈ closure (s ∩ Set.Iic a), (s ∩ Set.Iic a).Nonempty ∧ sSup (s ∩ Set.Iic a) = a,
    ∃ t ⊆ s, t.Nonempty ∧ BddAbove t ∧ sSup t = a,
    ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a,
    ∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ Ordinal.sup f = a].TFAE
-- Ordinal.mem_closure_iff_sup
theorem mem_closure_iff_sup :
    a ∈ closure s ↔
      ∃ (ι : Type u) (_ : Nonempty ι) (f : ι → Ordinal), (∀ i, f i ∈ s) ∧ sup.{u, u} f = a 
Arguments: {s : Set Ordinal.{u}} {a : Ordinal.{u}}
Goal: a ∈ closure s ↔ ∃ ι, ∃ (_ : Nonempty ι), ∃ f, (∀ (i : ι), f i ∈ s) ∧ Ordinal.sup f = a