ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- differentiableAt_pi''
@[fun_prop]
theorem differentiableAt_pi'' (hφ : ∀ i, DifferentiableAt 𝕜 (fun x => Φ x i) x) :
    DifferentiableAt 𝕜 Φ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} (hφ : ∀ (i : ι), DifferentiableAt 𝕜 (fun x => Φ x i) x)
Goal: DifferentiableAt 𝕜 Φ x
-- differentiableAt_apply
@[fun_prop]
theorem differentiableAt_apply (i : ι) (f : ∀ i, F' i) :
    DifferentiableAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (f : (i : ι) → F' i)
Goal: DifferentiableAt 𝕜 (fun f => f i) f
-- differentiableOn_pi
theorem differentiableOn_pi : DifferentiableOn 𝕜 Φ s ↔ ∀ i, DifferentiableOn 𝕜 (fun x => Φ x i) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: DifferentiableOn 𝕜 Φ s ↔ ∀ (i : ι), DifferentiableOn 𝕜 (fun x => Φ x i) s
-- differentiableOn_pi''
@[fun_prop]
theorem differentiableOn_pi'' (hφ : ∀ i, DifferentiableOn 𝕜 (fun x => Φ x i) s) :
    DifferentiableOn 𝕜 Φ s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} (hφ : ∀ (i : ι), DifferentiableOn 𝕜 (fun x => Φ x i) s)
Goal: DifferentiableOn 𝕜 Φ s
-- differentiableOn_apply
@[fun_prop]
theorem differentiableOn_apply (i : ι) (s' : Set (∀ i, F' i)) :
    DifferentiableOn (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) s' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (s' : Set ((i : ι) → F' i))
Goal: DifferentiableOn 𝕜 (fun f => f i) s'
-- differentiable_pi
theorem differentiable_pi : Differentiable 𝕜 Φ ↔ ∀ i, Differentiable 𝕜 fun x => Φ x i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: Differentiable 𝕜 Φ ↔ ∀ (i : ι), Differentiable 𝕜 fun x => Φ x i
-- differentiable_pi''
@[fun_prop]
theorem differentiable_pi'' (hφ : ∀ i, Differentiable 𝕜 fun x => Φ x i) :
    Differentiable 𝕜 Φ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} (hφ : ∀ (i : ι), Differentiable 𝕜 fun x => Φ x i)
Goal: Differentiable 𝕜 Φ
-- differentiable_apply
@[fun_prop]
theorem differentiable_apply (i : ι) :
    Differentiable (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι)
Goal: Differentiable 𝕜 fun f => f i
-- fderivWithin_pi
theorem fderivWithin_pi (h : ∀ i, DifferentiableWithinAt 𝕜 (φ i) s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x i => φ i x) s x = pi fun i => fderivWithin 𝕜 (φ i) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} (h : ∀ (i : ι), DifferentiableWithinAt 𝕜 (φ i) s x) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x i => φ i x) s x = ContinuousLinearMap.pi fun i => fderivWithin 𝕜 (φ i) s x
-- fderiv_pi
theorem fderiv_pi (h : ∀ i, DifferentiableAt 𝕜 (φ i) x) :
    fderiv 𝕜 (fun x i => φ i x) x = pi fun i => fderiv 𝕜 (φ i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} (h : ∀ (i : ι), DifferentiableAt 𝕜 (φ i) x)
Goal: fderiv 𝕜 (fun x i => φ i x) x = ContinuousLinearMap.pi fun i => fderiv 𝕜 (φ i) x
-- hasFDerivWithinAt_closure_of_tendsto_fderiv
theorem hasFDerivWithinAt_closure_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
    (f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
    (f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
    (h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
    HasFDerivWithinAt f f' (closure s) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F} (f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s) (f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y) (h : Filter.Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f'))
Goal: HasFDerivWithinAt f f' (closure s) x
-- hasDerivWithinAt_Ici_of_tendsto_deriv
theorem hasDerivWithinAt_Ici_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
    (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
    (f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E} (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a) (f_lim' : Filter.Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e))
Goal: HasDerivWithinAt f e (Set.Ici a) a
-- hasDerivWithinAt_Iic_of_tendsto_deriv
theorem hasDerivWithinAt_Iic_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ}
    {f : ℝ → E} (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a)
    (hs : s ∈ 𝓝[<] a) (f_lim' : Tendsto (fun x => deriv f x) (𝓝[<] a) (𝓝 e)) :
    HasDerivWithinAt f e (Iic a) a 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E} (f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[<] a) (f_lim' : Filter.Tendsto (fun x => deriv f x) (𝓝[<] a) (𝓝 e))
Goal: HasDerivWithinAt f e (Set.Iic a) a
-- hasDerivAt_of_hasDerivAt_of_ne
theorem hasDerivAt_of_hasDerivAt_of_ne {f g : ℝ → E} {x : ℝ}
    (f_diff : ∀ y ≠ x, HasDerivAt f (g y) y) (hf : ContinuousAt f x)
    (hg : ContinuousAt g x) : HasDerivAt f (g x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : ℝ → E} {x : ℝ} (f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y) (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: HasDerivAt f (g x) x
-- hasDerivAt_of_hasDerivAt_of_ne'
theorem hasDerivAt_of_hasDerivAt_of_ne' {f g : ℝ → E} {x : ℝ}
    (f_diff : ∀ y ≠ x, HasDerivAt f (g y) y) (hf : ContinuousAt f x)
    (hg : ContinuousAt g x) (y : ℝ) : HasDerivAt f (g y) y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : ℝ → E} {x : ℝ} (f_diff : ∀ (y : ℝ), y ≠ x → HasDerivAt f (g y) y) (hf : ContinuousAt f x) (hg : ContinuousAt g x) (y : ℝ)
Goal: HasDerivAt f (g y) y
-- iteratedDerivWithin_univ
theorem iteratedDerivWithin_univ : iteratedDerivWithin n f univ = iteratedDeriv n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedDerivWithin n f Set.univ = iteratedDeriv n f
-- iteratedDerivWithin_eq_iteratedFDerivWithin
theorem iteratedDerivWithin_eq_iteratedFDerivWithin : iteratedDerivWithin n f s x =
    (iteratedFDerivWithin 𝕜 n f s x : (Fin n → 𝕜) → F) fun _ : Fin n => 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜}
Goal: iteratedDerivWithin n f s x = (iteratedFDerivWithin 𝕜 n f s x) fun x => 1
-- iteratedDerivWithin_eq_equiv_comp
theorem iteratedDerivWithin_eq_equiv_comp : iteratedDerivWithin n f s =
    (ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDerivWithin 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜}
Goal: iteratedDerivWithin n f s = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDerivWithin 𝕜 n f s
-- iteratedFDerivWithin_eq_equiv_comp
theorem iteratedFDerivWithin_eq_equiv_comp :
    iteratedFDerivWithin 𝕜 n f s =
      ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDerivWithin n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜}
Goal: iteratedFDerivWithin 𝕜 n f s = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ iteratedDerivWithin n f s
-- iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod
theorem iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod {m : Fin n → 𝕜} :
    (iteratedFDerivWithin 𝕜 n f s x : (Fin n → 𝕜) → F) m =
      (∏ i, m i) • iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} {m : Fin n → 𝕜}
Goal: (iteratedFDerivWithin 𝕜 n f s x) m = (∏ i : Fin n, m i) • iteratedDerivWithin n f s x
-- norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin
theorem norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin :
    ‖iteratedFDerivWithin 𝕜 n f s x‖ = ‖iteratedDerivWithin n f s x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜}
Goal: ‖iteratedFDerivWithin 𝕜 n f s x‖ = ‖iteratedDerivWithin n f s x‖
-- iteratedDerivWithin_zero
@[simp]
theorem iteratedDerivWithin_zero : iteratedDerivWithin 0 f s = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜}
Goal: iteratedDerivWithin 0 f s = f
-- iteratedDerivWithin_one
@[simp]
theorem iteratedDerivWithin_one {x : 𝕜} (h : UniqueDiffWithinAt 𝕜 s x) :
    iteratedDerivWithin 1 f s x = derivWithin f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} (h : UniqueDiffWithinAt 𝕜 s x)
Goal: iteratedDerivWithin 1 f s x = derivWithin f s x
-- contDiffOn_of_continuousOn_differentiableOn_deriv
theorem contDiffOn_of_continuousOn_differentiableOn_deriv {n : ℕ∞}
    (Hcont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s)
    (Hdiff : ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s) :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞} (Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s) (Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s)
Goal: ContDiffOn 𝕜 n f s
-- contDiffOn_of_differentiableOn_deriv
theorem contDiffOn_of_differentiableOn_deriv {n : ℕ∞}
    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞} (h : ∀ (m : ℕ), ↑m ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s)
Goal: ContDiffOn 𝕜 n f s
-- ContDiffOn.continuousOn_iteratedDerivWithin
theorem ContDiffOn.continuousOn_iteratedDerivWithin {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s)
    (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s) : ContinuousOn (iteratedDerivWithin m f s) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s) (hmn : ↑m ≤ n) (hs : UniqueDiffOn 𝕜 s)
Goal: ContinuousOn (iteratedDerivWithin m f s) s
-- ContDiffWithinAt.differentiableWithinAt_iteratedDerivWithin
theorem ContDiffWithinAt.differentiableWithinAt_iteratedDerivWithin {n : ℕ∞} {m : ℕ}
    (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
    DifferentiableWithinAt 𝕜 (iteratedDerivWithin m f s) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} {n : ℕ∞} {m : ℕ} (h : ContDiffWithinAt 𝕜 n f s x) (hmn : ↑m < n) (hs : UniqueDiffOn 𝕜 (insert x s))
Goal: DifferentiableWithinAt 𝕜 (iteratedDerivWithin m f s) s x
-- ContDiffOn.differentiableOn_iteratedDerivWithin
theorem ContDiffOn.differentiableOn_iteratedDerivWithin {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s)
    (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 s) :
    DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞} {m : ℕ} (h : ContDiffOn 𝕜 n f s) (hmn : ↑m < n) (hs : UniqueDiffOn 𝕜 s)
Goal: DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s
-- contDiffOn_iff_continuousOn_differentiableOn_deriv
theorem contDiffOn_iff_continuousOn_differentiableOn_deriv {n : ℕ∞} (hs : UniqueDiffOn 𝕜 s) :
    ContDiffOn 𝕜 n f s ↔ (∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
      ∀ m : ℕ, (m : ℕ∞) < n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞} (hs : UniqueDiffOn 𝕜 s)
Goal: ContDiffOn 𝕜 n f s ↔
  (∀ (m : ℕ), ↑m ≤ n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
    ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s
-- iteratedDerivWithin_succ
theorem iteratedDerivWithin_succ {x : 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) :
    iteratedDerivWithin (n + 1) f s x = derivWithin (iteratedDerivWithin n f s) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: iteratedDerivWithin (n + 1) f s x = derivWithin (iteratedDerivWithin n f s) s x
-- iteratedDerivWithin_eq_iterate
theorem iteratedDerivWithin_eq_iterate {x : 𝕜} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedDerivWithin n f s x = (fun g : 𝕜 → F => derivWithin g s)^[n] f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedDerivWithin n f s x = (fun g => derivWithin g s)^[n] f x
-- iteratedDerivWithin_succ'
theorem iteratedDerivWithin_succ' {x : 𝕜} (hxs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedDerivWithin (n + 1) f s x = (iteratedDerivWithin n (derivWithin f s) s) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} (hxs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedDerivWithin (n + 1) f s x = iteratedDerivWithin n (derivWithin f s) s x
-- iteratedDeriv_eq_iteratedFDeriv
theorem iteratedDeriv_eq_iteratedFDeriv :
    iteratedDeriv n f x = (iteratedFDeriv 𝕜 n f x : (Fin n → 𝕜) → F) fun _ : Fin n => 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {x : 𝕜}
Goal: iteratedDeriv n f x = (iteratedFDeriv 𝕜 n f x) fun x => 1
-- iteratedDeriv_eq_equiv_comp
theorem iteratedDeriv_eq_equiv_comp : iteratedDeriv n f =
    (ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedDeriv n f = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f
-- iteratedFDeriv_eq_equiv_comp
theorem iteratedFDeriv_eq_equiv_comp : iteratedFDeriv 𝕜 n f =
    ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDeriv n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedFDeriv 𝕜 n f = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F) ∘ iteratedDeriv n f
-- iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod
theorem iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod {m : Fin n → 𝕜} :
    (iteratedFDeriv 𝕜 n f x : (Fin n → 𝕜) → F) m = (∏ i, m i) • iteratedDeriv n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {x : 𝕜} {m : Fin n → 𝕜}
Goal: (iteratedFDeriv 𝕜 n f x) m = (∏ i : Fin n, m i) • iteratedDeriv n f x
-- norm_iteratedFDeriv_eq_norm_iteratedDeriv
theorem norm_iteratedFDeriv_eq_norm_iteratedDeriv :
    ‖iteratedFDeriv 𝕜 n f x‖ = ‖iteratedDeriv n f x‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} {x : 𝕜}
Goal: ‖iteratedFDeriv 𝕜 n f x‖ = ‖iteratedDeriv n f x‖
-- iteratedDeriv_zero
@[simp]
theorem iteratedDeriv_zero : iteratedDeriv 0 f = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F}
Goal: iteratedDeriv 0 f = f
-- iteratedDeriv_one
@[simp]
theorem iteratedDeriv_one : iteratedDeriv 1 f = deriv f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F}
Goal: iteratedDeriv 1 f = deriv f
-- contDiff_iff_iteratedDeriv
theorem contDiff_iff_iteratedDeriv {n : ℕ∞} : ContDiff 𝕜 n f ↔
    (∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous (iteratedDeriv m f)) ∧
      ∀ m : ℕ, (m : ℕ∞) < n → Differentiable 𝕜 (iteratedDeriv m f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {n : ℕ∞}
Goal: ContDiff 𝕜 n f ↔
  (∀ (m : ℕ), ↑m ≤ n → Continuous (iteratedDeriv m f)) ∧ ∀ (m : ℕ), ↑m < n → Differentiable 𝕜 (iteratedDeriv m f)
-- contDiff_of_differentiable_iteratedDeriv
theorem contDiff_of_differentiable_iteratedDeriv {n : ℕ∞}
    (h : ∀ m : ℕ, (m : ℕ∞) ≤ n → Differentiable 𝕜 (iteratedDeriv m f)) : ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {n : ℕ∞} (h : ∀ (m : ℕ), ↑m ≤ n → Differentiable 𝕜 (iteratedDeriv m f))
Goal: ContDiff 𝕜 n f
-- ContDiff.continuous_iteratedDeriv
theorem ContDiff.continuous_iteratedDeriv {n : ℕ∞} (m : ℕ) (h : ContDiff 𝕜 n f)
    (hmn : (m : ℕ∞) ≤ n) : Continuous (iteratedDeriv m f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {n : ℕ∞} (m : ℕ) (h : ContDiff 𝕜 n f) (hmn : ↑m ≤ n)
Goal: Continuous (iteratedDeriv m f)
-- ContDiff.differentiable_iteratedDeriv
theorem ContDiff.differentiable_iteratedDeriv {n : ℕ∞} (m : ℕ) (h : ContDiff 𝕜 n f)
    (hmn : (m : ℕ∞) < n) : Differentiable 𝕜 (iteratedDeriv m f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {n : ℕ∞} (m : ℕ) (h : ContDiff 𝕜 n f) (hmn : ↑m < n)
Goal: Differentiable 𝕜 (iteratedDeriv m f)
-- iteratedDeriv_succ
theorem iteratedDeriv_succ : iteratedDeriv (n + 1) f = deriv (iteratedDeriv n f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedDeriv (n + 1) f = deriv (iteratedDeriv n f)
-- iteratedDeriv_eq_iterate
theorem iteratedDeriv_eq_iterate : iteratedDeriv n f = deriv^[n] f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedDeriv n f = deriv^[n] f
-- iteratedDeriv_succ'
theorem iteratedDeriv_succ' : iteratedDeriv (n + 1) f = iteratedDeriv n (deriv f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F}
Goal: iteratedDeriv (n + 1) f = iteratedDeriv n (deriv f)
-- Polynomial.IsUnitTrinomial.irreducible_of_coprime'
theorem irreducible_of_coprime' (hp : IsUnitTrinomial p)
    (h : ∀ z : ℂ, ¬(aeval z p = 0 ∧ aeval z (mirror p) = 0)) : Irreducible p 
Arguments: {p : Polynomial ℤ} (hp : p.IsUnitTrinomial) (h : ∀ (z : ℂ), ¬((Polynomial.aeval z) p = 0 ∧ (Polynomial.aeval z) p.mirror = 0))
Goal: Irreducible p
-- LipschitzWith.ae_lineDifferentiableAt
theorem ae_lineDifferentiableAt (hf : LipschitzWith C f) (v : E) :
    ∀ᵐ p ∂μ, LineDifferentiableAt ℝ f p v 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) (v : E)
Goal: ∀ᵐ (p : E) ∂μ, LineDifferentiableAt ℝ f p v
-- LipschitzWith.memℒp_lineDeriv
theorem memℒp_lineDeriv (hf : LipschitzWith C f) (v : E) :
    Memℒp (fun x ↦ lineDeriv ℝ f x v) ∞ μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} (hf : LipschitzWith C f) (v : E)
Goal: MeasureTheory.Memℒp (fun x => lineDeriv ℝ f x v) ⊤ μ
-- LipschitzWith.locallyIntegrable_lineDeriv
theorem locallyIntegrable_lineDeriv (hf : LipschitzWith C f) (v : E) :
    LocallyIntegrable (fun x ↦ lineDeriv ℝ f x v) μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) (v : E)
Goal: MeasureTheory.LocallyIntegrable (fun x => lineDeriv ℝ f x v) μ
-- LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
    (hf : LipschitzWith C f) (hg : Integrable g μ) (v : E) :
    Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
      (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f g : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) (hg : MeasureTheory.Integrable g μ) (v : E)
Goal: Filter.Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
  (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
-- LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
    (hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E) :
    Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
      (𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C : ℝ≥0} {f g : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) (h'f : HasCompactSupport f) (hg : Continuous g) (v : E)
Goal: Filter.Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0)
  (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))
-- LipschitzWith.integral_lineDeriv_mul_eq
theorem integral_lineDeriv_mul_eq
    (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E) :
    ∫ x, lineDeriv ℝ f x v * g x ∂μ = ∫ x, lineDeriv ℝ g x (-v) * f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {C D : ℝ≥0} {f g : E → ℝ} {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] (hf : LipschitzWith C f) (hg : LipschitzWith D g) (h'g : HasCompactSupport g) (v : E)
Goal: ∫ (x : E), lineDeriv ℝ f x v * g x ∂μ = ∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ