ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffWithinAt.congr
theorem ContDiffWithinAt.congr (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y)
    (hx : f₁ x = f x) : ContDiffWithinAt 𝕜 n f₁ 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 f₁ : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x)
Goal: ContDiffWithinAt 𝕜 n f₁ s x
-- ContDiffWithinAt.congr'
theorem ContDiffWithinAt.congr' (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y)
    (hx : x ∈ s) : ContDiffWithinAt 𝕜 n f₁ 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 f₁ : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : x ∈ s)
Goal: ContDiffWithinAt 𝕜 n f₁ s x
-- ContDiffWithinAt.mono_of_mem
theorem ContDiffWithinAt.mono_of_mem (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E}
    (hst : s ∈ 𝓝[t] x) : ContDiffWithinAt 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 : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : s ∈ 𝓝[t] x)
Goal: ContDiffWithinAt 𝕜 n f t x
-- ContDiffWithinAt.mono
theorem ContDiffWithinAt.mono (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : t ⊆ s) :
    ContDiffWithinAt 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 : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : t ⊆ s)
Goal: ContDiffWithinAt 𝕜 n f t x
-- ContDiffWithinAt.congr_nhds
theorem ContDiffWithinAt.congr_nhds (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E}
    (hst : 𝓝[s] x = 𝓝[t] x) : ContDiffWithinAt 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 : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) {t : Set E} (hst : 𝓝[s] x = 𝓝[t] x)
Goal: ContDiffWithinAt 𝕜 n f t x
-- contDiffWithinAt_congr_nhds
theorem contDiffWithinAt_congr_nhds {t : Set E} (hst : 𝓝[s] x = 𝓝[t] x) :
    ContDiffWithinAt 𝕜 n f s x ↔ ContDiffWithinAt 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 : Set E} {f : E → F} {x : E} {n : ℕ∞} {t : Set E} (hst : 𝓝[s] x = 𝓝[t] x)
Goal: ContDiffWithinAt 𝕜 n f s x ↔ ContDiffWithinAt 𝕜 n f t x
-- contDiffWithinAt_inter'
theorem contDiffWithinAt_inter' (h : t ∈ 𝓝[s] x) :
    ContDiffWithinAt 𝕜 n f (s ∩ t) 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 t : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : t ∈ 𝓝[s] x)
Goal: ContDiffWithinAt 𝕜 n f (s ∩ t) x ↔ ContDiffWithinAt 𝕜 n f s x
-- contDiffWithinAt_inter
theorem contDiffWithinAt_inter (h : t ∈ 𝓝 x) :
    ContDiffWithinAt 𝕜 n f (s ∩ t) 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 t : Set E} {f : E → F} {x : E} {n : ℕ∞} (h : t ∈ 𝓝 x)
Goal: ContDiffWithinAt 𝕜 n f (s ∩ t) x ↔ ContDiffWithinAt 𝕜 n f s x
-- contDiffWithinAt_insert_self
theorem contDiffWithinAt_insert_self :
    ContDiffWithinAt 𝕜 n f (insert x s) 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∞}
Goal: ContDiffWithinAt 𝕜 n f (insert x s) x ↔ ContDiffWithinAt 𝕜 n f s x
-- contDiffWithinAt_insert
theorem contDiffWithinAt_insert {y : E} :
    ContDiffWithinAt 𝕜 n f (insert y s) 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∞} {y : E}
Goal: ContDiffWithinAt 𝕜 n f (insert y s) x ↔ ContDiffWithinAt 𝕜 n f s x
-- ContDiffWithinAt.insert
protected theorem ContDiffWithinAt.insert (h : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n f (insert 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} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n f (insert x s) x
-- ContDiffWithinAt.differentiable_within_at'
theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n) :
    DifferentiableWithinAt 𝕜 f (insert 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} {x : E} {n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n)
Goal: DifferentiableWithinAt 𝕜 f (insert x s) x
-- ContDiffWithinAt.differentiableWithinAt
theorem ContDiffWithinAt.differentiableWithinAt (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n) :
    DifferentiableWithinAt 𝕜 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 : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n)
Goal: DifferentiableWithinAt 𝕜 f s x
-- contDiffWithinAt_succ_iff_hasFDerivWithinAt
theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : ℕ} :
    ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ∃ f' : E → E →L[𝕜] F,
      (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u 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}
Goal: ContDiffWithinAt 𝕜 (↑(n + 1)) f s x ↔
  ∃ u ∈ 𝓝[insert x s] x, ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 (↑n) f' u x
-- contDiffWithinAt_succ_iff_hasFDerivWithinAt'
theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : ℕ} :
    ContDiffWithinAt 𝕜 (n + 1 : ℕ) f s x ↔
      ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ∃ f' : E → E →L[𝕜] F,
        (∀ x ∈ u, HasFDerivWithinAt f (f' x) s 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}
Goal: ContDiffWithinAt 𝕜 (↑(n + 1)) f s x ↔
  ∃ u ∈ 𝓝[insert x s] x,
    u ⊆ insert x s ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) s x) ∧ ContDiffWithinAt 𝕜 (↑n) f' s x
-- HasFTaylorSeriesUpToOn.contDiffOn
theorem HasFTaylorSeriesUpToOn.contDiffOn {f' : E → FormalMultilinearSeries 𝕜 E F}
    (hf : HasFTaylorSeriesUpToOn n f f' 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∞} {f' : E → FormalMultilinearSeries 𝕜 E F} (hf : HasFTaylorSeriesUpToOn n f f' s)
Goal: ContDiffOn 𝕜 n f s
-- ContDiffOn.contDiffWithinAt
theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
    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 : ContDiffOn 𝕜 n f s) (hx : x ∈ s)
Goal: ContDiffWithinAt 𝕜 n f s x
-- ContDiffWithinAt.contDiffOn'
theorem ContDiffWithinAt.contDiffOn' {m : ℕ} (hm : (m : ℕ∞) ≤ n)
    (h : ContDiffWithinAt 𝕜 n f s x) :
    ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x 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} {n : ℕ∞} {m : ℕ} (hm : ↑m ≤ n) (h : ContDiffWithinAt 𝕜 n f s x)
Goal: ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 (↑m) f (insert x s ∩ u)
-- ContDiffWithinAt.contDiffOn
theorem ContDiffWithinAt.contDiffOn {m : ℕ} (hm : (m : ℕ∞) ≤ n) (h : ContDiffWithinAt 𝕜 n f s x) :
    ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ 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] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} {m : ℕ} (hm : ↑m ≤ n) (h : ContDiffWithinAt 𝕜 n f s x)
Goal: ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ContDiffOn 𝕜 (↑m) f u
-- ContDiffWithinAt.eventually
protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) :
    ∀ᶠ y in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y 
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)
Goal: ∀ᶠ (y : E) in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 (↑n) f s y
-- ContDiffOn.of_le
theorem ContDiffOn.of_le (h : ContDiffOn 𝕜 n f s) (hmn : m ≤ n) : ContDiffOn 𝕜 m 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} {m n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (hmn : m ≤ n)
Goal: ContDiffOn 𝕜 m f s
-- ContDiffOn.of_succ
theorem ContDiffOn.of_succ {n : ℕ} (h : ContDiffOn 𝕜 (n + 1) f 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 : ContDiffOn 𝕜 (↑n + 1) f s)
Goal: ContDiffOn 𝕜 (↑n) f s
-- ContDiffOn.one_of_succ
theorem ContDiffOn.one_of_succ {n : ℕ} (h : ContDiffOn 𝕜 (n + 1) f s) : ContDiffOn 𝕜 1 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 : ContDiffOn 𝕜 (↑n + 1) f s)
Goal: ContDiffOn 𝕜 1 f s
-- contDiffOn_iff_forall_nat_le
theorem contDiffOn_iff_forall_nat_le : ContDiffOn 𝕜 n f s ↔ ∀ m : ℕ, ↑m ≤ n → ContDiffOn 𝕜 m 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∞}
Goal: ContDiffOn 𝕜 n f s ↔ ∀ (m : ℕ), ↑m ≤ n → ContDiffOn 𝕜 (↑m) f s
-- contDiffOn_top
theorem contDiffOn_top : ContDiffOn 𝕜 ∞ f s ↔ ∀ n : ℕ, 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}
Goal: ContDiffOn 𝕜 ⊤ f s ↔ ∀ (n : ℕ), ContDiffOn 𝕜 (↑n) f s
-- contDiffOn_all_iff_nat
theorem contDiffOn_all_iff_nat : (∀ n, ContDiffOn 𝕜 n f s) ↔ ∀ n : ℕ, 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}
Goal: (∀ (n : ℕ∞), ContDiffOn 𝕜 n f s) ↔ ∀ (n : ℕ), ContDiffOn 𝕜 (↑n) f s
-- ContDiffOn.continuousOn
theorem ContDiffOn.continuousOn (h : ContDiffOn 𝕜 n 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} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s)
Goal: ContinuousOn f s
-- ContDiffOn.congr
theorem ContDiffOn.congr (h : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s, f₁ x = f x) :
    ContDiffOn 𝕜 n f₁ 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} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s, f₁ x = f x)
Goal: ContDiffOn 𝕜 n f₁ s
-- contDiffOn_congr
theorem contDiffOn_congr (h₁ : ∀ x ∈ s, f₁ x = f x) : ContDiffOn 𝕜 n f₁ 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 f₁ : E → F} {n : ℕ∞} (h₁ : ∀ x ∈ s, f₁ x = f x)
Goal: ContDiffOn 𝕜 n f₁ s ↔ ContDiffOn 𝕜 n f s
-- ContDiffOn.mono
theorem ContDiffOn.mono (h : ContDiffOn 𝕜 n f s) {t : Set E} (hst : t ⊆ s) : ContDiffOn 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 : Set E} {f : E → F} {n : ℕ∞} (h : ContDiffOn 𝕜 n f s) {t : Set E} (hst : t ⊆ s)
Goal: ContDiffOn 𝕜 n f t
-- ContDiffOn.congr_mono
theorem ContDiffOn.congr_mono (hf : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s₁, f₁ x = f x) (hs : s₁ ⊆ s) :
    ContDiffOn 𝕜 n f₁ s₁ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s s₁ : Set E} {f f₁ : E → F} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s₁, f₁ x = f x) (hs : s₁ ⊆ s)
Goal: ContDiffOn 𝕜 n f₁ s₁
-- ContDiffOn.differentiableOn
theorem ContDiffOn.differentiableOn (h : ContDiffOn 𝕜 n f s) (hn : 1 ≤ n) :
    DifferentiableOn 𝕜 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 : ContDiffOn 𝕜 n f s) (hn : 1 ≤ n)
Goal: DifferentiableOn 𝕜 f s
-- contDiffOn_of_locally_contDiffOn
theorem contDiffOn_of_locally_contDiffOn
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)) : 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 : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u))
Goal: ContDiffOn 𝕜 n f s
-- contDiffOn_succ_iff_hasFDerivWithinAt
theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
    ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
      ∀ x ∈ s, ∃ u ∈ 𝓝[insert x s] x, ∃ f' : E → E →L[𝕜] F,
        (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' 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} {n : ℕ}
Goal: ContDiffOn 𝕜 (↑(n + 1)) f s ↔
  ∀ x ∈ s, ∃ u ∈ 𝓝[insert x s] x, ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 (↑n) f' u
-- iteratedFDerivWithin_zero_apply
@[simp]
theorem iteratedFDerivWithin_zero_apply (m : Fin 0 → E) :
    (iteratedFDerivWithin 𝕜 0 f s 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] {s : Set E} {f : E → F} {x : E} (m : Fin 0 → E)
Goal: (iteratedFDerivWithin 𝕜 0 f s x) m = f x
-- iteratedFDerivWithin_zero_eq_comp
theorem iteratedFDerivWithin_zero_eq_comp :
    iteratedFDerivWithin 𝕜 0 f s = (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] {s : Set E} {f : E → F}
Goal: iteratedFDerivWithin 𝕜 0 f s = ⇑(continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f
-- norm_iteratedFDerivWithin_zero
@[simp]
theorem norm_iteratedFDerivWithin_zero : ‖iteratedFDerivWithin 𝕜 0 f s 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] {s : Set E} {f : E → F} {x : E}
Goal: ‖iteratedFDerivWithin 𝕜 0 f s x‖ = ‖f x‖
-- iteratedFDerivWithin_succ_apply_left
theorem iteratedFDerivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s 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] {s : Set E} {f : E → F} {x : E} {n : ℕ} (m : Fin (n + 1) → E)
Goal: (iteratedFDerivWithin 𝕜 (n + 1) f s x) m = ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x) (m 0)) (Fin.tail m)
-- iteratedFDerivWithin_succ_eq_comp_left
theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
    iteratedFDerivWithin 𝕜 (n + 1) f s =
      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F :
          (E →L[𝕜] (E [×n]→L[𝕜] F)) → (E [×n.succ]→L[𝕜] F)) ∘
        fderivWithin 𝕜 (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 : E → F} {n : ℕ}
Goal: iteratedFDerivWithin 𝕜 (n + 1) f s =
  ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s
-- fderivWithin_iteratedFDerivWithin
theorem fderivWithin_iteratedFDerivWithin {s : Set E} {n : ℕ} :
    fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s =
      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
        iteratedFDerivWithin 𝕜 (n + 1) f 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} {s : Set E} {n : ℕ}
Goal: fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s =
  ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F).symm ∘ iteratedFDerivWithin 𝕜 (n + 1) f s
-- norm_fderivWithin_iteratedFDerivWithin
theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
    ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
      ‖iteratedFDerivWithin 𝕜 (n + 1) 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}
Goal: ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ = ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖
-- iteratedFDerivWithin_succ_apply_right
theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
    (m : Fin (n + 1) → E) :
    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
      iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s 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] {s : Set E} {f : E → F} {x : E} {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (m : Fin (n + 1) → E)
Goal: (iteratedFDerivWithin 𝕜 (n + 1) f s x) m =
  ((iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x) (Fin.init m)) (m (Fin.last n))
-- iteratedFDerivWithin_succ_eq_comp_right
theorem iteratedFDerivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 (n + 1) f s x =
      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘
          iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) 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} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 (n + 1) f s x =
  (⇑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) x
-- norm_iteratedFDerivWithin_fderivWithin
theorem norm_iteratedFDerivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ =
      ‖iteratedFDerivWithin 𝕜 (n + 1) 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} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ = ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖
-- iteratedFDerivWithin_one_apply
@[simp]
theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
    iteratedFDerivWithin 𝕜 1 f s x m = fderivWithin 𝕜 f s 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] {s : Set E} {f : E → F} {x : E} (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E)
Goal: (iteratedFDerivWithin 𝕜 1 f s x) m = (fderivWithin 𝕜 f s x) (m 0)
-- iteratedFDerivWithin_two_apply
lemma iteratedFDerivWithin_two_apply (f : E → F) {z : E} (hs : UniqueDiffOn 𝕜 s) (hz : z ∈ s)
    (m : Fin 2 → E) :
    iteratedFDerivWithin 𝕜 2 f s z m = fderivWithin 𝕜 (fderivWithin 𝕜 f s) s 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] {s : Set E} (f : E → F) {z : E} (hs : UniqueDiffOn 𝕜 s) (hz : z ∈ s) (m : Fin 2 → E)
Goal: (iteratedFDerivWithin 𝕜 2 f s z) m = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s z) (m 0)) (m 1)
-- Filter.EventuallyEq.iteratedFDerivWithin'
theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] 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 f₁ : E → F} {x : E} (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t
-- Filter.EventuallyEq.iteratedFDerivWithin
protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 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 f₁ : E → F} {x : E} (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
-- Filter.EventuallyEq.iteratedFDerivWithin_eq
theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
    (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s 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 : Set E} {f f₁ : E → F} {x : E} (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x
-- iteratedFDerivWithin_congr
theorem iteratedFDerivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
    iteratedFDerivWithin 𝕜 n f₁ s 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 : Set E} {f f₁ : E → F} {x : E} (hs : Set.EqOn f₁ f s) (hx : x ∈ s) (n : ℕ)
Goal: iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x