ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAtFilter_iff_tendsto
theorem hasDerivAtFilter_iff_tendsto :
    HasDerivAtFilter f f' x L ↔
      Tendsto (fun x' : 𝕜 => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) L (𝓝 0) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜}
Goal: HasDerivAtFilter f f' x L ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) L (𝓝 0)
-- hasDerivWithinAt_iff_isLittleO
theorem hasDerivWithinAt_iff_isLittleO :
    HasDerivWithinAt f f' s x ↔
      (fun x' : 𝕜 => f x' - f x - (x' - x) • f') =o[𝓝[s] x] fun x' => x' - x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜}
Goal: HasDerivWithinAt f f' s x ↔ (fun x' => f x' - f x - (x' - x) • f') =o[𝓝[s] x] fun x' => x' - x
-- hasDerivWithinAt_iff_tendsto
theorem hasDerivWithinAt_iff_tendsto :
    HasDerivWithinAt f f' s x ↔
      Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝[s] x) (𝓝 0) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜}
Goal: HasDerivWithinAt f f' s x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝[s] x) (𝓝 0)
-- hasDerivAt_iff_isLittleO
theorem hasDerivAt_iff_isLittleO :
    HasDerivAt f f' x ↔ (fun x' : 𝕜 => f x' - f x - (x' - x) • f') =o[𝓝 x] fun x' => x' - x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivAt f f' x ↔ (fun x' => f x' - f x - (x' - x) • f') =o[𝓝 x] fun x' => x' - x
-- hasDerivAt_iff_tendsto
theorem hasDerivAt_iff_tendsto :
    HasDerivAt f f' x ↔ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝 x) (𝓝 0) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivAt f f' x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝 x) (𝓝 0)
-- HasDerivAtFilter.isBigO_sub
theorem HasDerivAtFilter.isBigO_sub (h : HasDerivAtFilter f f' x L) :
    (fun x' => f x' - f x) =O[L] fun x' => x' - x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (h : HasDerivAtFilter f f' x L)
Goal: (fun x' => f x' - f x) =O[L] fun x' => x' - x
-- HasDerivAtFilter.isBigO_sub_rev
nonrec theorem HasDerivAtFilter.isBigO_sub_rev (hf : HasDerivAtFilter f f' x L) (hf' : f' ≠ 0) :
    (fun x' => x' - x) =O[L] fun x' => f x' - f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (hf : HasDerivAtFilter f f' x L) (hf' : f' ≠ 0)
Goal: (fun x' => x' - x) =O[L] fun x' => f x' - f x
-- HasStrictDerivAt.hasDerivAt
theorem HasStrictDerivAt.hasDerivAt (h : HasStrictDerivAt f f' x) : HasDerivAt f f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasStrictDerivAt f f' x)
Goal: HasDerivAt f f' x
-- hasDerivWithinAt_congr_set'
theorem hasDerivWithinAt_congr_set' {s t : Set 𝕜} (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    HasDerivWithinAt f f' s x ↔ HasDerivWithinAt f f' t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: HasDerivWithinAt f f' s x ↔ HasDerivWithinAt f f' t x
-- hasDerivWithinAt_congr_set
theorem hasDerivWithinAt_congr_set {s t : Set 𝕜} (h : s =ᶠ[𝓝 x] t) :
    HasDerivWithinAt f f' s x ↔ HasDerivWithinAt f f' t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : s =ᶠ[𝓝 x] t)
Goal: HasDerivWithinAt f f' s x ↔ HasDerivWithinAt f f' t x
-- hasDerivWithinAt_diff_singleton
@[simp]
theorem hasDerivWithinAt_diff_singleton :
    HasDerivWithinAt f f' (s \ {x}) x ↔ HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜}
Goal: HasDerivWithinAt f f' (s \ {x}) x ↔ HasDerivWithinAt f f' s x
-- hasDerivWithinAt_Ioi_iff_Ici
@[simp]
theorem hasDerivWithinAt_Ioi_iff_Ici [PartialOrder 𝕜] :
    HasDerivWithinAt f f' (Ioi x) x ↔ HasDerivWithinAt f f' (Ici x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} [PartialOrder 𝕜]
Goal: HasDerivWithinAt f f' (Set.Ioi x) x ↔ HasDerivWithinAt f f' (Set.Ici x) x
-- hasDerivWithinAt_Iio_iff_Iic
@[simp]
theorem hasDerivWithinAt_Iio_iff_Iic [PartialOrder 𝕜] :
    HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} [PartialOrder 𝕜]
Goal: HasDerivWithinAt f f' (Set.Iio x) x ↔ HasDerivWithinAt f f' (Set.Iic x) x
-- HasDerivWithinAt.Ioi_iff_Ioo
theorem HasDerivWithinAt.Ioi_iff_Ioo [LinearOrder 𝕜] [OrderClosedTopology 𝕜] {x y : 𝕜} (h : x < y) :
    HasDerivWithinAt f f' (Ioo x y) x ↔ HasDerivWithinAt f f' (Ioi x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} [LinearOrder 𝕜] [OrderClosedTopology 𝕜] {x y : 𝕜} (h : x < y)
Goal: HasDerivWithinAt f f' (Set.Ioo x y) x ↔ HasDerivWithinAt f f' (Set.Ioi x) x
-- hasDerivAt_iff_isLittleO_nhds_zero
theorem hasDerivAt_iff_isLittleO_nhds_zero :
    HasDerivAt f f' x ↔ (fun h => f (x + h) - f x - h • f') =o[𝓝 0] fun h => h 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivAt f f' x ↔ (fun h => f (x + h) - f x - h • f') =o[𝓝 0] fun h => h
-- HasDerivAtFilter.mono
theorem HasDerivAtFilter.mono (h : HasDerivAtFilter f f' x L₂) (hst : L₁ ≤ L₂) :
    HasDerivAtFilter f f' x L₁ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L₁ L₂ : Filter 𝕜} (h : HasDerivAtFilter f f' x L₂) (hst : L₁ ≤ L₂)
Goal: HasDerivAtFilter f f' x L₁
-- HasDerivWithinAt.mono
theorem HasDerivWithinAt.mono (h : HasDerivWithinAt f f' t x) (hst : s ⊆ t) :
    HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : HasDerivWithinAt f f' t x) (hst : s ⊆ t)
Goal: HasDerivWithinAt f f' s x
-- HasDerivWithinAt.mono_of_mem
theorem HasDerivWithinAt.mono_of_mem (h : HasDerivWithinAt f f' t x) (hst : t ∈ 𝓝[s] x) :
    HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : HasDerivWithinAt f f' t x) (hst : t ∈ 𝓝[s] x)
Goal: HasDerivWithinAt f f' s x
-- HasDerivAt.hasDerivAtFilter
theorem HasDerivAt.hasDerivAtFilter (h : HasDerivAt f f' x) (hL : L ≤ 𝓝 x) :
    HasDerivAtFilter f f' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (h : HasDerivAt f f' x) (hL : L ≤ 𝓝 x)
Goal: HasDerivAtFilter f f' x L
-- HasDerivAt.hasDerivWithinAt
theorem HasDerivAt.hasDerivWithinAt (h : HasDerivAt f f' x) : HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivAt f f' x)
Goal: HasDerivWithinAt f f' s x
-- HasDerivWithinAt.differentiableWithinAt
theorem HasDerivWithinAt.differentiableWithinAt (h : HasDerivWithinAt f f' s x) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- HasDerivAt.differentiableAt
theorem HasDerivAt.differentiableAt (h : HasDerivAt f f' x) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x)
Goal: DifferentiableAt 𝕜 f x
-- hasDerivWithinAt_univ
@[simp]
theorem hasDerivWithinAt_univ : HasDerivWithinAt f f' univ x ↔ HasDerivAt f f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜}
Goal: HasDerivWithinAt f f' Set.univ x ↔ HasDerivAt f f' x
-- HasDerivAt.unique
theorem HasDerivAt.unique (h₀ : HasDerivAt f f₀' x) (h₁ : HasDerivAt f f₁' x) : f₀' = f₁' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f₀' f₁' : F} {x : 𝕜} (h₀ : HasDerivAt f f₀' x) (h₁ : HasDerivAt f f₁' x)
Goal: f₀' = f₁'
-- hasDerivWithinAt_inter'
theorem hasDerivWithinAt_inter' (h : t ∈ 𝓝[s] x) :
    HasDerivWithinAt f f' (s ∩ t) x ↔ HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : t ∈ 𝓝[s] x)
Goal: HasDerivWithinAt f f' (s ∩ t) x ↔ HasDerivWithinAt f f' s x
-- hasDerivWithinAt_inter
theorem hasDerivWithinAt_inter (h : t ∈ 𝓝 x) :
    HasDerivWithinAt f f' (s ∩ t) x ↔ HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : t ∈ 𝓝 x)
Goal: HasDerivWithinAt f f' (s ∩ t) x ↔ HasDerivWithinAt f f' s x
-- HasDerivWithinAt.union
theorem HasDerivWithinAt.union (hs : HasDerivWithinAt f f' s x) (ht : HasDerivWithinAt f f' t x) :
    HasDerivWithinAt f f' (s ∪ t) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (hs : HasDerivWithinAt f f' s x) (ht : HasDerivWithinAt f f' t x)
Goal: HasDerivWithinAt f f' (s ∪ t) x
-- HasDerivWithinAt.hasDerivAt
theorem HasDerivWithinAt.hasDerivAt (h : HasDerivWithinAt f f' s x) (hs : s ∈ 𝓝 x) :
    HasDerivAt f f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (hs : s ∈ 𝓝 x)
Goal: HasDerivAt f f' x
-- DifferentiableWithinAt.hasDerivWithinAt
theorem DifferentiableWithinAt.hasDerivWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    HasDerivWithinAt f (derivWithin f s x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : DifferentiableWithinAt 𝕜 f s x)
Goal: HasDerivWithinAt f (derivWithin f s x) s x
-- DifferentiableAt.hasDerivAt
theorem DifferentiableAt.hasDerivAt (h : DifferentiableAt 𝕜 f x) : HasDerivAt f (deriv f x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (h : DifferentiableAt 𝕜 f x)
Goal: HasDerivAt f (deriv f x) x
-- hasDerivAt_deriv_iff
@[simp]
theorem hasDerivAt_deriv_iff : HasDerivAt f (deriv f x) x ↔ DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}
Goal: HasDerivAt f (deriv f x) x ↔ DifferentiableAt 𝕜 f x
-- hasDerivWithinAt_derivWithin_iff
@[simp]
theorem hasDerivWithinAt_derivWithin_iff :
    HasDerivWithinAt f (derivWithin f s x) s x ↔ DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜}
Goal: HasDerivWithinAt f (derivWithin f s x) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableOn.hasDerivAt
theorem DifferentiableOn.hasDerivAt (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x) :
    HasDerivAt f (deriv f x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x)
Goal: HasDerivAt f (deriv f x) x
-- HasDerivAt.deriv
theorem HasDerivAt.deriv (h : HasDerivAt f f' x) : deriv f x = f' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x)
Goal: deriv f x = f'
-- deriv_eq
theorem deriv_eq {f' : 𝕜 → F} (h : ∀ x, HasDerivAt f (f' x) x) : deriv f = f' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f' : 𝕜 → F} (h : ∀ (x : 𝕜), HasDerivAt f (f' x) x)
Goal: deriv f = f'
-- HasDerivWithinAt.derivWithin
theorem HasDerivWithinAt.derivWithin (h : HasDerivWithinAt f f' s x)
    (hxs : UniqueDiffWithinAt 𝕜 s x) : derivWithin f s x = f' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin f s x = f'
-- fderivWithin_derivWithin
theorem fderivWithin_derivWithin : (fderivWithin 𝕜 f s x : 𝕜 → F) 1 = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜}
Goal: (fderivWithin 𝕜 f s x) 1 = derivWithin f s x
-- derivWithin_fderivWithin
theorem derivWithin_fderivWithin :
    smulRight (1 : 𝕜 →L[𝕜] 𝕜) (derivWithin f s x) = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜}
Goal: ContinuousLinearMap.smulRight 1 (derivWithin f s x) = fderivWithin 𝕜 f s x
-- norm_derivWithin_eq_norm_fderivWithin
theorem norm_derivWithin_eq_norm_fderivWithin : ‖derivWithin f s x‖ = ‖fderivWithin 𝕜 f s x‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜}
Goal: ‖derivWithin f s x‖ = ‖fderivWithin 𝕜 f s x‖
-- fderiv_deriv
theorem fderiv_deriv : (fderiv 𝕜 f x : 𝕜 → F) 1 = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}
Goal: (fderiv 𝕜 f x) 1 = deriv f x
-- fderiv_eq_smul_deriv
@[simp]
theorem fderiv_eq_smul_deriv (y : 𝕜) : (fderiv 𝕜 f x : 𝕜 → F) y = y • deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (y : 𝕜)
Goal: (fderiv 𝕜 f x) y = y • deriv f x
-- deriv_fderiv
theorem deriv_fderiv : smulRight (1 : 𝕜 →L[𝕜] 𝕜) (deriv f x) = fderiv 𝕜 f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}
Goal: ContinuousLinearMap.smulRight 1 (deriv f x) = fderiv 𝕜 f x
-- fderiv_eq_deriv_mul
lemma fderiv_eq_deriv_mul {f : 𝕜 → 𝕜} {x y : 𝕜} : (fderiv 𝕜 f x : 𝕜 → 𝕜) y = (deriv f x) * y 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {f : 𝕜 → 𝕜} {x y : 𝕜}
Goal: (fderiv 𝕜 f x) y = deriv f x * y
-- norm_deriv_eq_norm_fderiv
theorem norm_deriv_eq_norm_fderiv : ‖deriv f x‖ = ‖fderiv 𝕜 f x‖ 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}
Goal: ‖deriv f x‖ = ‖fderiv 𝕜 f x‖
-- DifferentiableAt.derivWithin
theorem DifferentiableAt.derivWithin (h : DifferentiableAt 𝕜 f x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin f s x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : DifferentiableAt 𝕜 f x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin f s x = deriv f x
-- HasDerivWithinAt.deriv_eq_zero
theorem HasDerivWithinAt.deriv_eq_zero (hd : HasDerivWithinAt f 0 s x)
    (H : UniqueDiffWithinAt 𝕜 s x) : deriv f x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hd : HasDerivWithinAt f 0 s x) (H : UniqueDiffWithinAt 𝕜 s x)
Goal: deriv f x = 0
-- derivWithin_of_mem
theorem derivWithin_of_mem (st : t ∈ 𝓝[s] x) (ht : UniqueDiffWithinAt 𝕜 s x)
    (h : DifferentiableWithinAt 𝕜 f t x) : derivWithin f s x = derivWithin f t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s t : Set 𝕜} (st : t ∈ 𝓝[s] x) (ht : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f t x)
Goal: derivWithin f s x = derivWithin f t x
-- derivWithin_subset
theorem derivWithin_subset (st : s ⊆ t) (ht : UniqueDiffWithinAt 𝕜 s x)
    (h : DifferentiableWithinAt 𝕜 f t x) : derivWithin f s x = derivWithin f t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s t : Set 𝕜} (st : s ⊆ t) (ht : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f t x)
Goal: derivWithin f s x = derivWithin f t x
-- derivWithin_congr_set'
theorem derivWithin_congr_set' (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    derivWithin f s x = derivWithin f t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s t : Set 𝕜} (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: derivWithin f s x = derivWithin f t x
-- derivWithin_congr_set
theorem derivWithin_congr_set (h : s =ᶠ[𝓝 x] t) : derivWithin f s x = derivWithin f t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s t : Set 𝕜} (h : s =ᶠ[𝓝 x] t)
Goal: derivWithin f s x = derivWithin f t x
-- derivWithin_univ
@[simp]
theorem derivWithin_univ : derivWithin f univ = deriv f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F}
Goal: derivWithin f Set.univ = deriv f
-- derivWithin_inter
theorem derivWithin_inter (ht : t ∈ 𝓝 x) : derivWithin f (s ∩ t) x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s t : Set 𝕜} (ht : t ∈ 𝓝 x)
Goal: derivWithin f (s ∩ t) x = derivWithin f s x
-- derivWithin_of_mem_nhds
theorem derivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : derivWithin f s x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (h : s ∈ 𝓝 x)
Goal: derivWithin f s x = deriv f x
-- derivWithin_of_isOpen
theorem derivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) : derivWithin f s x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hs : IsOpen s) (hx : x ∈ s)
Goal: derivWithin f s x = deriv f x
-- deriv_eqOn
lemma deriv_eqOn {f' : 𝕜 → F} (hs : IsOpen s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
    s.EqOn (deriv f) f' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {f' : 𝕜 → F} (hs : IsOpen s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
Goal: Set.EqOn (deriv f) f' s
-- deriv_mem_iff
theorem deriv_mem_iff {f : 𝕜 → F} {s : Set F} {x : 𝕜} :
    deriv f x ∈ s ↔
      DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ (0 : F) ∈ s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set F} {x : 𝕜}
Goal: deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s
-- derivWithin_mem_iff
theorem derivWithin_mem_iff {f : 𝕜 → F} {t : Set 𝕜} {s : Set F} {x : 𝕜} :
    derivWithin f t x ∈ s ↔
      DifferentiableWithinAt 𝕜 f t x ∧ derivWithin f t x ∈ s ∨
        ¬DifferentiableWithinAt 𝕜 f t x ∧ (0 : F) ∈ s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {t : Set 𝕜} {s : Set F} {x : 𝕜}
Goal: derivWithin f t x ∈ s ↔ DifferentiableWithinAt 𝕜 f t x ∧ derivWithin f t x ∈ s ∨ ¬DifferentiableWithinAt 𝕜 f t x ∧ 0 ∈ s
-- differentiableWithinAt_Ioi_iff_Ici
theorem differentiableWithinAt_Ioi_iff_Ici [PartialOrder 𝕜] :
    DifferentiableWithinAt 𝕜 f (Ioi x) x ↔ DifferentiableWithinAt 𝕜 f (Ici x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} [PartialOrder 𝕜]
Goal: DifferentiableWithinAt 𝕜 f (Set.Ioi x) x ↔ DifferentiableWithinAt 𝕜 f (Set.Ici x) x
-- derivWithin_Ioi_eq_Ici
theorem derivWithin_Ioi_eq_Ici {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E)
    (x : ℝ) : derivWithin f (Ioi x) x = derivWithin f (Ici x) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (x : ℝ)
Goal: derivWithin f (Set.Ioi x) x = derivWithin f (Set.Ici x) x
-- Filter.EventuallyEq.hasDerivAtFilter_iff
theorem Filter.EventuallyEq.hasDerivAtFilter_iff (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x)
    (h₁ : f₀' = f₁') : HasDerivAtFilter f₀ f₀' x L ↔ HasDerivAtFilter f₁ f₁' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₀ f₁ : 𝕜 → F} {f₀' f₁' : F} {x : 𝕜} {L : Filter 𝕜} (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : f₀' = f₁')
Goal: HasDerivAtFilter f₀ f₀' x L ↔ HasDerivAtFilter f₁ f₁' x L
-- HasDerivAtFilter.congr_of_eventuallyEq
theorem HasDerivAtFilter.congr_of_eventuallyEq (h : HasDerivAtFilter f f' x L) (hL : f₁ =ᶠ[L] f)
    (hx : f₁ x = f x) : HasDerivAtFilter f₁ f' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (h : HasDerivAtFilter f f' x L) (hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x)
Goal: HasDerivAtFilter f₁ f' x L
-- HasDerivWithinAt.congr_mono
theorem HasDerivWithinAt.congr_mono (h : HasDerivWithinAt f f' s x) (ht : ∀ x ∈ t, f₁ x = f x)
    (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasDerivWithinAt f₁ f' t x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s t : Set 𝕜} (h : HasDerivWithinAt f f' s x) (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: HasDerivWithinAt f₁ f' t x
-- HasDerivWithinAt.congr
theorem HasDerivWithinAt.congr (h : HasDerivWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x)
    (hx : f₁ x = f x) : HasDerivWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: HasDerivWithinAt f₁ f' s x
-- HasDerivWithinAt.congr_of_mem
theorem HasDerivWithinAt.congr_of_mem (h : HasDerivWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x)
    (hx : x ∈ s) : HasDerivWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x) (hx : x ∈ s)
Goal: HasDerivWithinAt f₁ f' s x
-- HasDerivWithinAt.congr_of_eventuallyEq
theorem HasDerivWithinAt.congr_of_eventuallyEq (h : HasDerivWithinAt f f' s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasDerivWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasDerivWithinAt f₁ f' s x
-- Filter.EventuallyEq.hasDerivWithinAt_iff
theorem Filter.EventuallyEq.hasDerivWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    HasDerivWithinAt f₁ f' s x ↔ HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasDerivWithinAt f₁ f' s x ↔ HasDerivWithinAt f f' s x