ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Arrow.mapAugmentedCechConerve_right
@[simps]
def mapAugmentedCechConerve {f g : Arrow C}
    [∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
    [∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
    f.augmentedCechConerve ⟶ g.augmentedCechConerve 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] [∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout g.left (fun x => g.right) fun x => g.hom] (F : f ⟶ g)
Goal: (CategoryTheory.Arrow.mapAugmentedCechConerve F).right = CategoryTheory.Arrow.mapCechConerve F
-- CategoryTheory.Arrow.mapAugmentedCechConerve_left
@[simps]
def mapAugmentedCechConerve {f g : Arrow C}
    [∀ n : ℕ, HasWidePushout f.left (fun _ : Fin (n + 1) => f.right) fun _ => f.hom]
    [∀ n : ℕ, HasWidePushout g.left (fun _ : Fin (n + 1) => g.right) fun _ => g.hom] (F : f ⟶ g) :
    f.augmentedCechConerve ⟶ g.augmentedCechConerve 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {f g : CategoryTheory.Arrow C} [∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] [∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout g.left (fun x => g.right) fun x => g.hom] (F : f ⟶ g)
Goal: (CategoryTheory.Arrow.mapAugmentedCechConerve F).left = F.left
-- CategoryTheory.CosimplicialObject.cechConerve_map
@[simps]
def cechConerve : Arrow C ⥤ CosimplicialObject C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom]
Goal: ∀ {X Y : CategoryTheory.Arrow C} (F : X ⟶ Y),
  CategoryTheory.CosimplicialObject.cechConerve.map F = CategoryTheory.Arrow.mapCechConerve F
-- CategoryTheory.CosimplicialObject.cechConerve_obj
@[simps]
def cechConerve : Arrow C ⥤ CosimplicialObject C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (f : CategoryTheory.Arrow C)
Goal: CategoryTheory.CosimplicialObject.cechConerve.obj f = f.cechConerve
-- CategoryTheory.CosimplicialObject.augmentedCechConerve_map
@[simps]
def augmentedCechConerve : Arrow C ⥤ CosimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom]
Goal: ∀ {X Y : CategoryTheory.Arrow C} (F : X ⟶ Y),
  CategoryTheory.CosimplicialObject.augmentedCechConerve.map F = CategoryTheory.Arrow.mapAugmentedCechConerve F
-- CategoryTheory.CosimplicialObject.augmentedCechConerve_obj
@[simps]
def augmentedCechConerve : Arrow C ⥤ CosimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (f : CategoryTheory.Arrow C)
Goal: CategoryTheory.CosimplicialObject.augmentedCechConerve.obj f = f.augmentedCechConerve
-- CategoryTheory.CosimplicialObject.equivalenceLeftToRight_right
@[simps]
def equivalenceLeftToRight (F : Arrow C) (X : CosimplicialObject.Augmented C)
    (G : F.augmentedCechConerve ⟶ X) : F ⟶ Augmented.toArrow.obj X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F.augmentedCechConerve ⟶ X)
Goal: (CategoryTheory.CosimplicialObject.equivalenceLeftToRight F X G).right =
  CategoryTheory.Limits.WidePushout.ι (fun x => F.hom) 0 ≫ G.right.app (SimplexCategory.mk 0)
-- CategoryTheory.CosimplicialObject.equivalenceLeftToRight_left
@[simps]
def equivalenceLeftToRight (F : Arrow C) (X : CosimplicialObject.Augmented C)
    (G : F.augmentedCechConerve ⟶ X) : F ⟶ Augmented.toArrow.obj X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F.augmentedCechConerve ⟶ X)
Goal: (CategoryTheory.CosimplicialObject.equivalenceLeftToRight F X G).left = G.left
-- CategoryTheory.CosimplicialObject.equivalenceRightToLeft_left
@[simps!]
def equivalenceRightToLeft (F : Arrow C) (X : CosimplicialObject.Augmented C)
    (G : F ⟶ Augmented.toArrow.obj X) : F.augmentedCechConerve ⟶ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F ⟶ CategoryTheory.CosimplicialObject.Augmented.toArrow.obj X)
Goal: (CategoryTheory.CosimplicialObject.equivalenceRightToLeft F X G).left = G.left
-- CategoryTheory.CosimplicialObject.equivalenceRightToLeft_right_app
@[simps!]
def equivalenceRightToLeft (F : Arrow C) (X : CosimplicialObject.Augmented C)
    (G : F ⟶ Augmented.toArrow.obj X) : F.augmentedCechConerve ⟶ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F ⟶ CategoryTheory.CosimplicialObject.Augmented.toArrow.obj X) (x : SimplexCategory)
Goal: (CategoryTheory.CosimplicialObject.equivalenceRightToLeft F X G).right.app x =
  CategoryTheory.Limits.WidePushout.desc (G.left ≫ X.hom.app x)
    (fun i => G.right ≫ X.right.map ((SimplexCategory.mk 0).const x i)) ⋯
-- CategoryTheory.CosimplicialObject.cechConerveEquiv_symm_apply
@[simps]
def cechConerveEquiv (F : Arrow C) (X : CosimplicialObject.Augmented C) :
    (F.augmentedCechConerve ⟶ X) ≃ (F ⟶ Augmented.toArrow.obj X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F ⟶ CategoryTheory.CosimplicialObject.Augmented.toArrow.obj X)
Goal: (CategoryTheory.CosimplicialObject.cechConerveEquiv F X).symm G =
  CategoryTheory.CosimplicialObject.equivalenceRightToLeft F X G
-- CategoryTheory.CosimplicialObject.cechConerveEquiv_apply
@[simps]
def cechConerveEquiv (F : Arrow C) (X : CosimplicialObject.Augmented C) :
    (F.augmentedCechConerve ⟶ X) ≃ (F ⟶ Augmented.toArrow.obj X) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom] (F : CategoryTheory.Arrow C) (X : CategoryTheory.CosimplicialObject.Augmented C) (G : F.augmentedCechConerve ⟶ X)
Goal: (CategoryTheory.CosimplicialObject.cechConerveEquiv F X) G =
  CategoryTheory.CosimplicialObject.equivalenceLeftToRight F X G
-- CategoryTheory.CechNerveTerminalFrom.wideCospan.limitIsoPi_inv_comp_pi
@[reassoc (attr := simp)]
lemma wideCospan.limitIsoPi_inv_comp_pi [Finite ι] (X : C) (j : ι) :
    (wideCospan.limitIsoPi ι X).inv ≫ WidePullback.π _ j = Pi.π _ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] (ι : Type w) [CategoryTheory.Limits.HasFiniteProducts C] [Finite ι] (X : C) (j : ι)
Goal: (CategoryTheory.CechNerveTerminalFrom.wideCospan.limitIsoPi ι X).inv ≫
    CategoryTheory.Limits.WidePullback.π (fun x => CategoryTheory.Limits.terminal.from X) j =
  CategoryTheory.Limits.Pi.π (fun x => X) j
-- CategoryTheory.CechNerveTerminalFrom.wideCospan.limitIsoPi_hom_comp_pi
@[reassoc (attr := simp)]
lemma wideCospan.limitIsoPi_hom_comp_pi [Finite ι] (X : C) (j : ι) :
    (wideCospan.limitIsoPi ι X).hom ≫ Pi.π _ j = WidePullback.π _ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasTerminal C] (ι : Type w) [CategoryTheory.Limits.HasFiniteProducts C] [Finite ι] (X : C) (j : ι)
Goal: (CategoryTheory.CechNerveTerminalFrom.wideCospan.limitIsoPi ι X).hom ≫ CategoryTheory.Limits.Pi.π (fun x => X) j =
  CategoryTheory.Limits.WidePullback.π (fun x => CategoryTheory.Limits.terminal.from X) j
-- HasFTaylorSeriesUpToOn.zero_eq
structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
  (s : Set E) : Prop where
  zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
    HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} {s : Set E} (self : HasFTaylorSeriesUpToOn n f p s) (x : E)
Goal: x ∈ s → (p x 0).uncurry0 = f x
-- HasFTaylorSeriesUpToOn.cont
structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
  (s : Set E) : Prop where
  zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
    HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} {s : Set E} (self : HasFTaylorSeriesUpToOn n f p s) (m : ℕ)
Goal: ↑m ≤ n → ContinuousOn (fun x => p x m) s
-- HasFTaylorSeriesUpToOn.fderivWithin
structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
  (s : Set E) : Prop where
  zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
    HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} {p : E → FormalMultilinearSeries 𝕜 E F} {s : Set E} (self : HasFTaylorSeriesUpToOn n f p s) (m : ℕ)
Goal: ↑m < n → ∀ x ∈ s, HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft s x
-- HasFTaylorSeriesUpToOn.zero_eq'
theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (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} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s)
Goal: p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x)
-- HasFTaylorSeriesUpToOn.congr
theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
    (h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (h₁ : ∀ x ∈ s, f₁ x = f x)
Goal: HasFTaylorSeriesUpToOn n f₁ p s
-- HasFTaylorSeriesUpToOn.mono
theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
    HasFTaylorSeriesUpToOn n f p 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s)
Goal: HasFTaylorSeriesUpToOn n f p t
-- HasFTaylorSeriesUpToOn.of_le
theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
    HasFTaylorSeriesUpToOn m f p 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n)
Goal: HasFTaylorSeriesUpToOn m f p s
-- HasFTaylorSeriesUpToOn.continuousOn
theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s)
Goal: ContinuousOn f s
-- hasFTaylorSeriesUpToOn_zero_iff
theorem hasFTaylorSeriesUpToOn_zero_iff :
    HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = 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} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x
-- hasFTaylorSeriesUpToOn_top_iff
theorem hasFTaylorSeriesUpToOn_top_iff :
    HasFTaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFTaylorSeriesUpToOn n f p 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} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpToOn ⊤ f p s ↔ ∀ (n : ℕ), HasFTaylorSeriesUpToOn (↑n) f p s
-- hasFTaylorSeriesUpToOn_top_iff'
theorem hasFTaylorSeriesUpToOn_top_iff' :
    HasFTaylorSeriesUpToOn ∞ f p s ↔
      (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
        ∀ m : ℕ, ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft 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} {p : E → FormalMultilinearSeries 𝕜 E F}
Goal: HasFTaylorSeriesUpToOn ⊤ f p s ↔
  (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧ ∀ (m : ℕ), ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x
-- HasFTaylorSeriesUpToOn.hasFDerivWithinAt
theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
    (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) (hx : x ∈ s)
Goal: HasFDerivWithinAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) s x
-- HasFTaylorSeriesUpToOn.differentiableOn
theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
Goal: DifferentiableOn 𝕜 f s
-- HasFTaylorSeriesUpToOn.hasFDerivAt
theorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
    (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x)
Goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) x
-- HasFTaylorSeriesUpToOn.eventually_hasFDerivAt
theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)
    (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
    ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) 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∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x)
Goal: ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) y
-- HasFTaylorSeriesUpToOn.differentiableAt
theorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
    (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {n : ℕ∞} {p : E → FormalMultilinearSeries 𝕜 E F} (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x)
Goal: DifferentiableAt 𝕜 f x
-- hasFTaylorSeriesUpToOn_succ_iff_left
theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
    HasFTaylorSeriesUpToOn (n + 1) f p s ↔
      HasFTaylorSeriesUpToOn n f p s ∧
        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
          ContinuousOn (fun x => p x (n + 1)) 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} {p : E → FormalMultilinearSeries 𝕜 E F} {n : ℕ}
Goal: HasFTaylorSeriesUpToOn (↑n + 1) f p s ↔
  HasFTaylorSeriesUpToOn (↑n) f p s ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧ ContinuousOn (fun x => p x (n + 1)) s
-- HasFTaylorSeriesUpToOn.shift_of_succ
theorem HasFTaylorSeriesUpToOn.shift_of_succ
    {n : ℕ} (H : HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s) :
    (HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
      (fun x => (p x).shift)) 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} {p : E → FormalMultilinearSeries 𝕜 E F} {n : ℕ} (H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s)
Goal: HasFTaylorSeriesUpToOn (↑n) (fun x => (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x => (p x).shift) s
-- hasFTaylorSeriesUpToOn_succ_iff_right
theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
    HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
      (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
          HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
            (fun x => (p x).shift) 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} {p : E → FormalMultilinearSeries 𝕜 E F} {n : ℕ}
Goal: HasFTaylorSeriesUpToOn (↑(n + 1)) f p s ↔
  (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
    (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
      HasFTaylorSeriesUpToOn (↑n) (fun x => (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x => (p x).shift) s
-- contDiffWithinAt_nat
theorem contDiffWithinAt_nat {n : ℕ} :
    ContDiffWithinAt 𝕜 n f s x ↔ ∃ u ∈ 𝓝[insert x s] x,
      ∃ p : E → FormalMultilinearSeries 𝕜 E F, HasFTaylorSeriesUpToOn n f p 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}
Goal: ContDiffWithinAt 𝕜 (↑n) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑n) f p u
-- ContDiffWithinAt.of_le
theorem ContDiffWithinAt.of_le (h : ContDiffWithinAt 𝕜 n f s x) (hmn : m ≤ n) :
    ContDiffWithinAt 𝕜 m 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} {m n : ℕ∞} (h : ContDiffWithinAt 𝕜 n f s x) (hmn : m ≤ n)
Goal: ContDiffWithinAt 𝕜 m f s x
-- contDiffWithinAt_iff_forall_nat_le
theorem contDiffWithinAt_iff_forall_nat_le :
    ContDiffWithinAt 𝕜 n f s x ↔ ∀ m : ℕ, ↑m ≤ n → ContDiffWithinAt 𝕜 m 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 s x ↔ ∀ (m : ℕ), ↑m ≤ n → ContDiffWithinAt 𝕜 (↑m) f s x
-- contDiffWithinAt_top
theorem contDiffWithinAt_top : ContDiffWithinAt 𝕜 ∞ f s x ↔ ∀ n : ℕ, 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}
Goal: ContDiffWithinAt 𝕜 ⊤ f s x ↔ ∀ (n : ℕ), ContDiffWithinAt 𝕜 (↑n) f s x
-- ContDiffWithinAt.continuousWithinAt
theorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt 𝕜 n f s x) :
    ContinuousWithinAt 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)
Goal: ContinuousWithinAt f s x
-- ContDiffWithinAt.congr_of_eventuallyEq
theorem ContDiffWithinAt.congr_of_eventuallyEq (h : ContDiffWithinAt 𝕜 n f s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (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 : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: ContDiffWithinAt 𝕜 n f₁ s x
-- ContDiffWithinAt.congr_of_eventuallyEq_insert
theorem ContDiffWithinAt.congr_of_eventuallyEq_insert (h : ContDiffWithinAt 𝕜 n f s x)
    (h₁ : f₁ =ᶠ[𝓝[insert x s] x] f) : 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 : f₁ =ᶠ[𝓝[insert x s] x] f)
Goal: ContDiffWithinAt 𝕜 n f₁ s x
-- ContDiffWithinAt.congr_of_eventually_eq'
theorem ContDiffWithinAt.congr_of_eventually_eq' (h : ContDiffWithinAt 𝕜 n f s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (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 : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s)
Goal: ContDiffWithinAt 𝕜 n f₁ s x
-- Filter.EventuallyEq.contDiffWithinAt_iff
theorem Filter.EventuallyEq.contDiffWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    ContDiffWithinAt 𝕜 n f₁ 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 f₁ : E → F} {x : E} {n : ℕ∞} (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: ContDiffWithinAt 𝕜 n f₁ s x ↔ ContDiffWithinAt 𝕜 n f s x