ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- RCLike.summable_conj
@[simp]
theorem summable_conj {f : α → 𝕜} : (Summable fun x => conj (f x)) ↔ Summable f 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜}
Goal: (Summable fun x => (starRingEnd 𝕜) (f x)) ↔ Summable f
-- RCLike.conj_tsum
theorem conj_tsum (f : α → 𝕜) : conj (∑' a, f a) = ∑' a, conj (f a) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] (f : α → 𝕜)
Goal: (starRingEnd 𝕜) (∑' (a : α), f a) = ∑' (a : α), (starRingEnd 𝕜) (f a)
-- RCLike.hasSum_ofReal
@[simp, norm_cast]
theorem hasSum_ofReal {f : α → ℝ} {x : ℝ} : HasSum (fun x => (f x : 𝕜)) x ↔ HasSum f x 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → ℝ} {x : ℝ}
Goal: HasSum (fun x => ↑(f x)) ↑x ↔ HasSum f x
-- RCLike.summable_ofReal
@[simp, norm_cast]
theorem summable_ofReal {f : α → ℝ} : (Summable fun x => (f x : 𝕜)) ↔ Summable f 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → ℝ}
Goal: (Summable fun x => ↑(f x)) ↔ Summable f
-- RCLike.ofReal_tsum
@[norm_cast]
theorem ofReal_tsum (f : α → ℝ) : (↑(∑' a, f a) : 𝕜) = ∑' a, (f a : 𝕜) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] (f : α → ℝ)
Goal: ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)
-- RCLike.hasSum_re
theorem hasSum_re {f : α → 𝕜} {x : 𝕜} (h : HasSum f x) : HasSum (fun x => re (f x)) (re x) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} (h : HasSum f x)
Goal: HasSum (fun x => RCLike.re (f x)) (RCLike.re x)
-- RCLike.hasSum_im
theorem hasSum_im {f : α → 𝕜} {x : 𝕜} (h : HasSum f x) : HasSum (fun x => im (f x)) (im x) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} {x : 𝕜} (h : HasSum f x)
Goal: HasSum (fun x => RCLike.im (f x)) (RCLike.im x)
-- RCLike.re_tsum
theorem re_tsum {f : α → 𝕜} (h : Summable f) : re (∑' a, f a) = ∑' a, re (f a) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} (h : Summable f)
Goal: RCLike.re (∑' (a : α), f a) = ∑' (a : α), RCLike.re (f a)
-- RCLike.im_tsum
theorem im_tsum {f : α → 𝕜} (h : Summable f) : im (∑' a, f a) = ∑' a, im (f a) 
Arguments: {α : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] {f : α → 𝕜} (h : Summable f)
Goal: RCLike.im (∑' (a : α), f a) = ∑' (a : α), RCLike.im (f a)
-- RCLike.hasSum_iff
theorem hasSum_iff (f : α → 𝕜) (c : 𝕜) :
    HasSum f c ↔ HasSum (fun x => re (f x)) (re c) ∧ HasSum (fun x => im (f x)) (im c) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] (f : α → 𝕜) (c : 𝕜)
Goal: HasSum f c ↔ HasSum (fun x => RCLike.re (f x)) (RCLike.re c) ∧ HasSum (fun x => RCLike.im (f x)) (RCLike.im c)
-- Complex.hasSum_conj
theorem hasSum_conj {f : α → ℂ} {x : ℂ} : HasSum (fun x => conj (f x)) x ↔ HasSum f (conj x) 
Arguments: {α : Type u_1} {f : α → ℂ} {x : ℂ}
Goal: HasSum (fun x => (starRingEnd ℂ) (f x)) x ↔ HasSum f ((starRingEnd ℂ) x)
-- Complex.hasSum_conj'
theorem hasSum_conj' {f : α → ℂ} {x : ℂ} : HasSum (fun x => conj (f x)) (conj x) ↔ HasSum f x 
Arguments: {α : Type u_1} {f : α → ℂ} {x : ℂ}
Goal: HasSum (fun x => (starRingEnd ℂ) (f x)) ((starRingEnd ℂ) x) ↔ HasSum f x
-- Complex.summable_conj
theorem summable_conj {f : α → ℂ} : (Summable fun x => conj (f x)) ↔ Summable f 
Arguments: {α : Type u_1} {f : α → ℂ}
Goal: (Summable fun x => (starRingEnd ℂ) (f x)) ↔ Summable f
-- Complex.conj_tsum
theorem conj_tsum (f : α → ℂ) : conj (∑' a, f a) = ∑' a, conj (f a) 
Arguments: {α : Type u_1} (f : α → ℂ)
Goal: (starRingEnd ℂ) (∑' (a : α), f a) = ∑' (a : α), (starRingEnd ℂ) (f a)
-- Complex.hasSum_ofReal
@[simp, norm_cast]
theorem hasSum_ofReal {f : α → ℝ} {x : ℝ} : HasSum (fun x => (f x : ℂ)) x ↔ HasSum f x 
Arguments: {α : Type u_1} {f : α → ℝ} {x : ℝ}
Goal: HasSum (fun x => ↑(f x)) ↑x ↔ HasSum f x
-- Complex.summable_ofReal
@[simp, norm_cast]
theorem summable_ofReal {f : α → ℝ} : (Summable fun x => (f x : ℂ)) ↔ Summable f 
Arguments: {α : Type u_1} {f : α → ℝ}
Goal: (Summable fun x => ↑(f x)) ↔ Summable f
-- Complex.ofReal_tsum
@[norm_cast]
theorem ofReal_tsum (f : α → ℝ) : (↑(∑' a, f a) : ℂ) = ∑' a, ↑(f a) 
Arguments: {α : Type u_1} (f : α → ℝ)
Goal: ↑(∑' (a : α), f a) = ∑' (a : α), ↑(f a)
-- Complex.hasSum_re
theorem hasSum_re {f : α → ℂ} {x : ℂ} (h : HasSum f x) : HasSum (fun x => (f x).re) x.re 
Arguments: {α : Type u_1} {f : α → ℂ} {x : ℂ} (h : HasSum f x)
Goal: HasSum (fun x => (f x).re) x.re
-- Complex.hasSum_im
theorem hasSum_im {f : α → ℂ} {x : ℂ} (h : HasSum f x) : HasSum (fun x => (f x).im) x.im 
Arguments: {α : Type u_1} {f : α → ℂ} {x : ℂ} (h : HasSum f x)
Goal: HasSum (fun x => (f x).im) x.im
-- Complex.re_tsum
theorem re_tsum {f : α → ℂ} (h : Summable f) : (∑' a, f a).re = ∑' a, (f a).re 
Arguments: {α : Type u_1} {f : α → ℂ} (h : Summable f)
Goal: (∑' (a : α), f a).re = ∑' (a : α), (f a).re
-- Complex.im_tsum
theorem im_tsum {f : α → ℂ} (h : Summable f) : (∑' a, f a).im = ∑' a, (f a).im 
Arguments: {α : Type u_1} {f : α → ℂ} (h : Summable f)
Goal: (∑' (a : α), f a).im = ∑' (a : α), (f a).im
-- Complex.hasSum_iff
theorem hasSum_iff (f : α → ℂ) (c : ℂ) :
    HasSum f c ↔ HasSum (fun x => (f x).re) c.re ∧ HasSum (fun x => (f x).im) c.im 
Arguments: {α : Type u_1} (f : α → ℂ) (c : ℂ)
Goal: HasSum f c ↔ HasSum (fun x => (f x).re) c.re ∧ HasSum (fun x => (f x).im) c.im
-- Complex.mem_slitPlane_iff
lemma mem_slitPlane_iff {z : ℂ} : z ∈ slitPlane ↔ 0 < z.re ∨ z.im ≠ 0 
Arguments: {z : ℂ}
Goal: z ∈ Complex.slitPlane ↔ 0 < z.re ∨ z.im ≠ 0
-- Complex.slitPlane_eq_union
lemma slitPlane_eq_union : slitPlane = {z | 0 < z.re} ∪ {z | z.im ≠ 0} 
Goal: Complex.slitPlane = {z | 0 < z.re} ∪ {z | z.im ≠ 0}
-- Complex.isOpen_slitPlane
lemma isOpen_slitPlane : IsOpen slitPlane 
Goal: IsOpen Complex.slitPlane
-- Complex.ofReal_mem_slitPlane
@[simp]
lemma ofReal_mem_slitPlane {x : ℝ} : ↑x ∈ slitPlane ↔ 0 < x 
Arguments: {x : ℝ}
Goal: ↑x ∈ Complex.slitPlane ↔ 0 < x
-- Complex.neg_ofReal_mem_slitPlane
@[simp]
lemma neg_ofReal_mem_slitPlane {x : ℝ} : -↑x ∈ slitPlane ↔ x < 0 
Arguments: {x : ℝ}
Goal: -↑x ∈ Complex.slitPlane ↔ x < 0
-- Complex.one_mem_slitPlane
@[simp] lemma one_mem_slitPlane : 1 ∈ slitPlane 
Goal: 1 ∈ Complex.slitPlane
-- Complex.zero_not_mem_slitPlane
@[simp]
lemma zero_not_mem_slitPlane : 0 ∉ slitPlane 
Goal: 0 ∉ Complex.slitPlane
-- Complex.natCast_mem_slitPlane
@[simp]
lemma natCast_mem_slitPlane {n : ℕ} : ↑n ∈ slitPlane ↔ n ≠ 0 
Arguments: {n : ℕ}
Goal: ↑n ∈ Complex.slitPlane ↔ n ≠ 0
-- Complex.ofNat_mem_slitPlane
@[simp]
lemma ofNat_mem_slitPlane (n : ℕ) [n.AtLeastTwo] : no_index (OfNat.ofNat n) ∈ slitPlane 
Arguments: (n : ℕ) [n.AtLeastTwo]
Goal: OfNat.ofNat n ∈ Complex.slitPlane
-- Complex.mem_slitPlane_iff_not_le_zero
lemma mem_slitPlane_iff_not_le_zero {z : ℂ} : z ∈ slitPlane ↔ ¬z ≤ 0 
Arguments: {z : ℂ}
Goal: z ∈ Complex.slitPlane ↔ ¬z ≤ 0
-- Complex.compl_Iic_zero
protected lemma compl_Iic_zero : (Set.Iic 0)ᶜ = slitPlane 
Goal: (Set.Iic 0)ᶜ = Complex.slitPlane
-- Complex.slitPlane_ne_zero
lemma slitPlane_ne_zero {z : ℂ} (hz : z ∈ slitPlane) : z ≠ 0 
Arguments: {z : ℂ} (hz : z ∈ Complex.slitPlane)
Goal: z ≠ 0
-- Complex.ball_one_subset_slitPlane
lemma ball_one_subset_slitPlane : Metric.ball 1 1 ⊆ slitPlane 
Goal: Metric.ball 1 1 ⊆ Complex.slitPlane
-- Complex.mem_slitPlane_of_norm_lt_one
lemma mem_slitPlane_of_norm_lt_one {z : ℂ} (hz : ‖z‖ < 1) : 1 + z ∈ slitPlane 
Arguments: {z : ℂ} (hz : ‖z‖ < 1)
Goal: 1 + z ∈ Complex.slitPlane
-- HasFDerivAtFilter.isLittleO
@[mk_iff hasFDerivAtFilter_iff_isLittleO]
structure HasFDerivAtFilter (f : E → F) (f' : E →L[𝕜] F) (x : E) (L : Filter E) : Prop where
  of_isLittleO :: isLittleO : (fun x' => f x' - f x - f' (x' - x)) =o[L] fun x' => x' - x
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (self : HasFDerivAtFilter f f' x L)
Goal: (fun x' => f x' - f x - f' (x' - x)) =o[L] fun x' => x' - x
-- hasFDerivAtFilter_iff_isLittleO
@[mk_iff hasFDerivAtFilter_iff_isLittleO]
structure HasFDerivAtFilter (f : E → F) (f' : E →L[𝕜] F) (x : E) (L : Filter E) : Prop where
  of_isLittleO :: isLittleO : (fun x' => f x' - f x - f' (x' - x)) =o[L] fun x' => x' - x
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) (f' : E →L[𝕜] F) (x : E) (L : Filter E)
Goal: HasFDerivAtFilter f f' x L ↔ (fun x' => f x' - f x - f' (x' - x)) =o[L] fun x' => x' - x
-- fderivWithin_def
irreducible_def fderivWithin (f : E → F) (s : Set E) (x : E) : E →L[𝕜] F 
Arguments: (𝕜 : Type u_6) [NontriviallyNormedField 𝕜] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) (s : Set E) (x : E)
Goal: fderivWithin 𝕜 f s x =
  if 𝓝[s \ {x}] x = ⊥ then 0 else if h : ∃ f', HasFDerivWithinAt f f' s x then Classical.choose h else 0
-- fderiv_def
irreducible_def fderiv (f : E → F) (x : E) : E →L[𝕜] F 
Arguments: (𝕜 : Type u_6) [NontriviallyNormedField 𝕜] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) (x : E)
Goal: fderiv 𝕜 f x = if h : ∃ f', HasFDerivAt f f' x then Classical.choose h else 0
-- fderivWithin_zero_of_isolated
theorem fderivWithin_zero_of_isolated (h : 𝓝[s \ {x}] x = ⊥) : fderivWithin 𝕜 f s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : 𝓝[s \ {x}] x = ⊥)
Goal: fderivWithin 𝕜 f s x = 0
-- fderivWithin_zero_of_nmem_closure
theorem fderivWithin_zero_of_nmem_closure (h : x ∉ closure s) : fderivWithin 𝕜 f s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : x ∉ closure s)
Goal: fderivWithin 𝕜 f s x = 0
-- fderivWithin_zero_of_not_differentiableWithinAt
theorem fderivWithin_zero_of_not_differentiableWithinAt (h : ¬DifferentiableWithinAt 𝕜 f s x) :
    fderivWithin 𝕜 f s x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : ¬DifferentiableWithinAt 𝕜 f s x)
Goal: fderivWithin 𝕜 f s x = 0
-- fderiv_zero_of_not_differentiableAt
theorem fderiv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : fderiv 𝕜 f x = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : ¬DifferentiableAt 𝕜 f x)
Goal: fderiv 𝕜 f x = 0
-- HasFDerivWithinAt.lim
theorem HasFDerivWithinAt.lim (h : HasFDerivWithinAt f f' s x) {α : Type*} (l : Filter α)
    {c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ n in l, x + d n ∈ s)
    (clim : Tendsto (fun n => ‖c n‖) l atTop) (cdlim : Tendsto (fun n => c n • d n) l (𝓝 v)) :
    Tendsto (fun n => c n • (f (x + d n) - f x)) l (𝓝 (f' v)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) {α : Type u_6} (l : Filter α) {c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ (n : α) in l, x + d n ∈ s) (clim : Filter.Tendsto (fun n => ‖c n‖) l Filter.atTop) (cdlim : Filter.Tendsto (fun n => c n • d n) l (𝓝 v))
Goal: Filter.Tendsto (fun n => c n • (f (x + d n) - f x)) l (𝓝 (f' v))
-- HasFDerivWithinAt.unique_on
theorem HasFDerivWithinAt.unique_on (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt f f₁' s x) : EqOn f' f₁' (tangentConeAt 𝕜 s x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' f₁' : E →L[𝕜] F} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt f f₁' s x)
Goal: Set.EqOn (⇑f') (⇑f₁') (tangentConeAt 𝕜 s x)
-- UniqueDiffWithinAt.eq
theorem UniqueDiffWithinAt.eq (H : UniqueDiffWithinAt 𝕜 s x) (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt f f₁' s x) : f' = f₁' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' f₁' : E →L[𝕜] F} {x : E} {s : Set E} (H : UniqueDiffWithinAt 𝕜 s x) (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt f f₁' s x)
Goal: f' = f₁'
-- UniqueDiffOn.eq
theorem UniqueDiffOn.eq (H : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : HasFDerivWithinAt f f' s x)
    (h₁ : HasFDerivWithinAt f f₁' s x) : f' = f₁' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' f₁' : E →L[𝕜] F} {x : E} {s : Set E} (H : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : HasFDerivWithinAt f f' s x) (h₁ : HasFDerivWithinAt f f₁' s x)
Goal: f' = f₁'
-- hasFDerivAtFilter_iff_tendsto
theorem hasFDerivAtFilter_iff_tendsto :
    HasFDerivAtFilter f f' x L ↔
      Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E}
Goal: HasFDerivAtFilter f f' x L ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0)
-- hasFDerivWithinAt_iff_tendsto
theorem hasFDerivWithinAt_iff_tendsto :
    HasFDerivWithinAt f f' s x ↔
      Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝[s] x) (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E}
Goal: HasFDerivWithinAt f f' s x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝[s] x) (𝓝 0)
-- hasFDerivAt_iff_tendsto
theorem hasFDerivAt_iff_tendsto :
    HasFDerivAt f f' x ↔ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝 x) (𝓝 0) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E}
Goal: HasFDerivAt f f' x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝 x) (𝓝 0)
-- hasFDerivAt_iff_isLittleO_nhds_zero
theorem hasFDerivAt_iff_isLittleO_nhds_zero :
    HasFDerivAt f f' x ↔ (fun h : E => f (x + h) - f x - f' h) =o[𝓝 0] fun h => h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E}
Goal: HasFDerivAt f f' x ↔ (fun h => f (x + h) - f x - f' h) =o[𝓝 0] fun h => h
-- HasFDerivAt.le_of_lip'
theorem HasFDerivAt.le_of_lip' {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀)
    {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀) {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖)
Goal: ‖f'‖ ≤ C
-- HasFDerivAt.le_of_lipschitzOn
theorem HasFDerivAt.le_of_lipschitzOn
    {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀)
    {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀) {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖f'‖ ≤ ↑C
-- HasFDerivAt.le_of_lipschitz
theorem HasFDerivAt.le_of_lipschitz {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀)
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀) {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖f'‖ ≤ ↑C
-- HasFDerivAtFilter.mono
nonrec theorem HasFDerivAtFilter.mono (h : HasFDerivAtFilter f f' x L₂) (hst : L₁ ≤ L₂) :
    HasFDerivAtFilter f f' x L₁ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L₁ L₂ : Filter E} (h : HasFDerivAtFilter f f' x L₂) (hst : L₁ ≤ L₂)
Goal: HasFDerivAtFilter f f' x L₁
-- HasFDerivWithinAt.mono_of_mem
theorem HasFDerivWithinAt.mono_of_mem (h : HasFDerivWithinAt f f' t x) (hst : t ∈ 𝓝[s] x) :
    HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' t x) (hst : t ∈ 𝓝[s] x)
Goal: HasFDerivWithinAt f f' s x
-- HasFDerivWithinAt.mono
nonrec theorem HasFDerivWithinAt.mono (h : HasFDerivWithinAt f f' t x) (hst : s ⊆ t) :
    HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' t x) (hst : s ⊆ t)
Goal: HasFDerivWithinAt f f' s x
-- HasFDerivAt.hasFDerivAtFilter
theorem HasFDerivAt.hasFDerivAtFilter (h : HasFDerivAt f f' x) (hL : L ≤ 𝓝 x) :
    HasFDerivAtFilter f f' x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (h : HasFDerivAt f f' x) (hL : L ≤ 𝓝 x)
Goal: HasFDerivAtFilter f f' x L
-- HasFDerivAt.hasFDerivWithinAt
@[fun_prop]
theorem HasFDerivAt.hasFDerivWithinAt (h : HasFDerivAt f f' x) : HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivAt f f' x)
Goal: HasFDerivWithinAt f f' s x
-- HasFDerivWithinAt.differentiableWithinAt
@[fun_prop]
theorem HasFDerivWithinAt.differentiableWithinAt (h : HasFDerivWithinAt f f' s x) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- HasFDerivAt.differentiableAt
@[fun_prop]
theorem HasFDerivAt.differentiableAt (h : HasFDerivAt f f' x) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x)
Goal: DifferentiableAt 𝕜 f x
-- hasFDerivWithinAt_univ
@[simp]
theorem hasFDerivWithinAt_univ : HasFDerivWithinAt f f' univ x ↔ HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E}
Goal: HasFDerivWithinAt f f' Set.univ x ↔ HasFDerivAt f f' x
-- hasFDerivWithinAt_of_mem_nhds
theorem hasFDerivWithinAt_of_mem_nhds (h : s ∈ 𝓝 x) :
    HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : s ∈ 𝓝 x)
Goal: HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x
-- hasFDerivWithinAt_of_isOpen
lemma hasFDerivWithinAt_of_isOpen (h : IsOpen s) (hx : x ∈ s) :
    HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : IsOpen s) (hx : x ∈ s)
Goal: HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x
-- hasFDerivWithinAt_insert
theorem hasFDerivWithinAt_insert {y : E} :
    HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} {y : E}
Goal: HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x
-- HasFDerivWithinAt.insert
protected theorem HasFDerivWithinAt.insert (h : HasFDerivWithinAt g g' s x) :
    HasFDerivWithinAt g g' (insert x s) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {g : E → F} {g' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt g g' s x)
Goal: HasFDerivWithinAt g g' (insert x s) x
-- hasFDerivWithinAt_diff_singleton
theorem hasFDerivWithinAt_diff_singleton (y : E) :
    HasFDerivWithinAt f f' (s \ {y}) x ↔ HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (y : E)
Goal: HasFDerivWithinAt f f' (s \ {y}) x ↔ HasFDerivWithinAt f f' s x
-- HasStrictFDerivAt.isBigO_sub
theorem HasStrictFDerivAt.isBigO_sub (hf : HasStrictFDerivAt f f' x) :
    (fun p : E × E => f p.1 - f p.2) =O[𝓝 (x, x)] fun p : E × E => p.1 - p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: (fun p => f p.1 - f p.2) =O[𝓝 (x, x)] fun p => p.1 - p.2
-- HasFDerivAtFilter.isBigO_sub
theorem HasFDerivAtFilter.isBigO_sub (h : HasFDerivAtFilter f f' x L) :
    (fun x' => f x' - f x) =O[L] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (h : HasFDerivAtFilter f f' x L)
Goal: (fun x' => f x' - f x) =O[L] fun x' => x' - x
-- HasStrictFDerivAt.hasFDerivAt
@[fun_prop]
protected theorem HasStrictFDerivAt.hasFDerivAt (hf : HasStrictFDerivAt f f' x) :
    HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasFDerivAt f f' x
-- HasStrictFDerivAt.differentiableAt
protected theorem HasStrictFDerivAt.differentiableAt (hf : HasStrictFDerivAt f f' x) :
    DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: DifferentiableAt 𝕜 f x