ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffAt.fderiv
protected theorem ContDiffAt.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞}
    (hf : ContDiffAt 𝕜 n (Function.uncurry f) (x₀, g x₀)) (hg : ContDiffAt 𝕜 m g x₀)
    (hmn : m + 1 ≤ n) : ContDiffAt 𝕜 m (fun x => fderiv 𝕜 (f x) (g x)) x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x₀ : E} {m : ℕ∞} {f : E → F → G} {g : E → F} {n : ℕ∞} (hf : ContDiffAt 𝕜 n (Function.uncurry f) (x₀, g x₀)) (hg : ContDiffAt 𝕜 m g x₀) (hmn : m + 1 ≤ n)
Goal: ContDiffAt 𝕜 m (fun x => fderiv 𝕜 (f x) (g x)) x₀
-- ContDiffAt.fderiv_right
theorem ContDiffAt.fderiv_right (hf : ContDiffAt 𝕜 n f x₀) (hmn : (m + 1 : ℕ∞) ≤ n) :
    ContDiffAt 𝕜 m (fderiv 𝕜 f) x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x₀ : E} {m n : ℕ∞} (hf : ContDiffAt 𝕜 n f x₀) (hmn : m + 1 ≤ n)
Goal: ContDiffAt 𝕜 m (fderiv 𝕜 f) x₀
-- ContDiffAt.iteratedFDeriv_right
theorem ContDiffAt.iteratedFDeriv_right {i : ℕ} (hf : ContDiffAt 𝕜 n f x₀)
    (hmn : (m + i : ℕ∞) ≤ n) : ContDiffAt 𝕜 m (iteratedFDeriv 𝕜 i f) x₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x₀ : E} {m n : ℕ∞} {i : ℕ} (hf : ContDiffAt 𝕜 n f x₀) (hmn : m + ↑i ≤ n)
Goal: ContDiffAt 𝕜 m (iteratedFDeriv 𝕜 i f) x₀
-- ContDiff.fderiv
protected theorem ContDiff.fderiv {f : E → F → G} {g : E → F} {n m : ℕ∞}
    (hf : ContDiff 𝕜 m <| Function.uncurry f) (hg : ContDiff 𝕜 n g) (hnm : n + 1 ≤ m) :
    ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F → G} {g : E → F} {n m : ℕ∞} (hf : ContDiff 𝕜 m (Function.uncurry f)) (hg : ContDiff 𝕜 n g) (hnm : n + 1 ≤ m)
Goal: ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x)
-- ContDiff.fderiv_right
theorem ContDiff.fderiv_right (hf : ContDiff 𝕜 n f) (hmn : (m + 1 : ℕ∞) ≤ n) :
    ContDiff 𝕜 m (fderiv 𝕜 f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {m n : ℕ∞} (hf : ContDiff 𝕜 n f) (hmn : m + 1 ≤ n)
Goal: ContDiff 𝕜 m (fderiv 𝕜 f)
-- ContDiff.iteratedFDeriv_right
theorem ContDiff.iteratedFDeriv_right {i : ℕ} (hf : ContDiff 𝕜 n f)
    (hmn : (m + i : ℕ∞) ≤ n) : ContDiff 𝕜 m (iteratedFDeriv 𝕜 i f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {m n : ℕ∞} {i : ℕ} (hf : ContDiff 𝕜 n f) (hmn : m + ↑i ≤ n)
Goal: ContDiff 𝕜 m (iteratedFDeriv 𝕜 i f)
-- Continuous.fderiv
theorem Continuous.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞}
    (hf : ContDiff 𝕜 n <| Function.uncurry f) (hg : Continuous g) (hn : 1 ≤ n) :
    Continuous fun x => fderiv 𝕜 (f x) (g x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F → G} {g : E → F} {n : ℕ∞} (hf : ContDiff 𝕜 n (Function.uncurry f)) (hg : Continuous g) (hn : 1 ≤ n)
Goal: Continuous fun x => fderiv 𝕜 (f x) (g x)
-- ContDiff.fderiv_apply
theorem ContDiff.fderiv_apply {f : E → F → G} {g k : E → F} {n m : ℕ∞}
    (hf : ContDiff 𝕜 m <| Function.uncurry f) (hg : ContDiff 𝕜 n g) (hk : ContDiff 𝕜 n k)
    (hnm : n + 1 ≤ m) : ContDiff 𝕜 n fun x => fderiv 𝕜 (f x) (g x) (k x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F → G} {g k : E → F} {n m : ℕ∞} (hf : ContDiff 𝕜 m (Function.uncurry f)) (hg : ContDiff 𝕜 n g) (hk : ContDiff 𝕜 n k) (hnm : n + 1 ≤ m)
Goal: ContDiff 𝕜 n fun x => (fderiv 𝕜 (f x) (g x)) (k x)
-- contDiffOn_fderivWithin_apply
theorem contDiffOn_fderivWithin_apply {m n : ℕ∞} {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s)
    (hs : UniqueDiffOn 𝕜 s) (hmn : m + 1 ≤ n) :
    ContDiffOn 𝕜 m (fun p : E × E => (fderivWithin 𝕜 f s p.1 : E →L[𝕜] F) p.2) (s ×ˢ univ) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m n : ℕ∞} {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hmn : m + 1 ≤ n)
Goal: ContDiffOn 𝕜 m (fun p => (fderivWithin 𝕜 f s p.1) p.2) (s ×ˢ Set.univ)
-- ContDiffOn.continuousOn_fderivWithin_apply
theorem ContDiffOn.continuousOn_fderivWithin_apply (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
    (hn : 1 ≤ n) :
    ContinuousOn (fun p : E × E => (fderivWithin 𝕜 f s p.1 : E → F) p.2) (s ×ˢ univ) 
Arguments: {𝕜 : Type u_1} [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∞} (hf : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) (hn : 1 ≤ n)
Goal: ContinuousOn (fun p => (fderivWithin 𝕜 f s p.1) p.2) (s ×ˢ Set.univ)
-- ContDiff.contDiff_fderiv_apply
theorem ContDiff.contDiff_fderiv_apply {f : E → F} (hf : ContDiff 𝕜 n f) (hmn : m + 1 ≤ n) :
    ContDiff 𝕜 m fun p : E × E => (fderiv 𝕜 f p.1 : E →L[𝕜] F) p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m n : ℕ∞} {f : E → F} (hf : ContDiff 𝕜 n f) (hmn : m + 1 ≤ n)
Goal: ContDiff 𝕜 m fun p => (fderiv 𝕜 f p.1) p.2
-- hasFTaylorSeriesUpToOn_pi
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} {p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)}
Goal: HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
  ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
-- hasFTaylorSeriesUpToOn_pi'
@[simp]
theorem hasFTaylorSeriesUpToOn_pi' :
    HasFTaylorSeriesUpToOn n Φ P' s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (fun x => Φ x i)
        (fun x m => (@ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _ i).compContinuousMultilinearMap
          (P' x m)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)}
Goal: HasFTaylorSeriesUpToOn n Φ P' s ↔
  ∀ (i : ι),
    HasFTaylorSeriesUpToOn n (fun x => Φ x i)
      (fun x m => (ContinuousLinearMap.proj i).compContinuousMultilinearMap (P' x m)) s
-- contDiffWithinAt_pi
theorem contDiffWithinAt_pi :
    ContDiffWithinAt 𝕜 n Φ s x ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => Φ x i) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: ContDiffWithinAt 𝕜 n Φ s x ↔ ∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x => Φ x i) s x
-- contDiffOn_pi
theorem contDiffOn_pi : ContDiffOn 𝕜 n Φ s ↔ ∀ i, ContDiffOn 𝕜 n (fun x => Φ x i) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: ContDiffOn 𝕜 n Φ s ↔ ∀ (i : ι), ContDiffOn 𝕜 n (fun x => Φ x i) s
-- contDiffAt_pi
theorem contDiffAt_pi : ContDiffAt 𝕜 n Φ x ↔ ∀ i, ContDiffAt 𝕜 n (fun x => Φ x i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: ContDiffAt 𝕜 n Φ x ↔ ∀ (i : ι), ContDiffAt 𝕜 n (fun x => Φ x i) x
-- contDiff_pi
theorem contDiff_pi : ContDiff 𝕜 n Φ ↔ ∀ i, ContDiff 𝕜 n fun x => Φ x i 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: ContDiff 𝕜 n Φ ↔ ∀ (i : ι), ContDiff 𝕜 n fun x => Φ x i
-- contDiff_update
theorem contDiff_update [DecidableEq ι] (k : ℕ∞) (x : ∀ i, F' i) (i : ι) :
    ContDiff 𝕜 k (update x i) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_3} [Fintype ι] {F' : ι → Type u_5} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] [DecidableEq ι] (k : ℕ∞) (x : (i : ι) → F' i) (i : ι)
Goal: ContDiff 𝕜 k (Function.update x i)
-- contDiff_single
theorem contDiff_single [DecidableEq ι] (k : ℕ∞) (i : ι) :
    ContDiff 𝕜 k (Pi.single i : F' i → ∀ i, F' i) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_3} [Fintype ι] (F' : ι → Type u_5) [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] [DecidableEq ι] (k : ℕ∞) (i : ι)
Goal: ContDiff 𝕜 k (Pi.single i)
-- contDiff_apply
theorem contDiff_apply (i : ι) : ContDiff 𝕜 n fun f : ι → E => f i 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type uE) [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {ι : Type u_3} [Fintype ι] (i : ι)
Goal: ContDiff 𝕜 n fun f => f i
-- contDiff_apply_apply
theorem contDiff_apply_apply (i : ι) (j : ι') : ContDiff 𝕜 n fun f : ι → ι' → E => f i j 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type uE) [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {ι : Type u_3} {ι' : Type u_4} [Fintype ι] [Fintype ι'] (i : ι) (j : ι')
Goal: ContDiff 𝕜 n fun f => f i j
-- HasFTaylorSeriesUpToOn.add
theorem HasFTaylorSeriesUpToOn.add {q g} (hf : HasFTaylorSeriesUpToOn n f p s)
    (hg : HasFTaylorSeriesUpToOn n g q s) : HasFTaylorSeriesUpToOn n (f + g) (p + q) s 
Arguments: {𝕜 : Type u_1} [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} {q : E → FormalMultilinearSeries 𝕜 E F} {g : E → F} (hf : HasFTaylorSeriesUpToOn n f p s) (hg : HasFTaylorSeriesUpToOn n g q s)
Goal: HasFTaylorSeriesUpToOn n (f + g) (p + q) s
-- contDiff_add
theorem contDiff_add : ContDiff 𝕜 n fun p : F × F => p.1 + p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞}
Goal: ContDiff 𝕜 n fun p => p.1 + p.2
-- ContDiffWithinAt.add
theorem ContDiffWithinAt.add {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x + g x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => f x + g x) s x
-- ContDiffAt.add
theorem ContDiffAt.add {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) :
    ContDiffAt 𝕜 n (fun x => f x + g x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x)
Goal: ContDiffAt 𝕜 n (fun x => f x + g x) x
-- ContDiff.add
theorem ContDiff.add {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n fun x => f x + g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n fun x => f x + g x
-- ContDiffOn.add
theorem ContDiffOn.add {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s)
    (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x + g x) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s)
Goal: ContDiffOn 𝕜 n (fun x => f x + g x) s
-- iteratedFDerivWithin_add_apply
theorem iteratedFDerivWithin_add_apply {f g : E → F} (hf : ContDiffOn 𝕜 i f s)
    (hg : ContDiffOn 𝕜 i g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 i (f + g) s x =
      iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} {i : ℕ} {f g : E → F} (hf : ContDiffOn 𝕜 (↑i) f s) (hg : ContDiffOn 𝕜 (↑i) g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 i (f + g) s x = iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x
-- iteratedFDerivWithin_add_apply'
theorem iteratedFDerivWithin_add_apply' {f g : E → F} (hf : ContDiffOn 𝕜 i f s)
    (hg : ContDiffOn 𝕜 i g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 i (fun x => f x + g x) s x =
      iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} {i : ℕ} {f g : E → F} (hf : ContDiffOn 𝕜 (↑i) f s) (hg : ContDiffOn 𝕜 (↑i) g s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 i (fun x => f x + g x) s x = iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x
-- iteratedFDeriv_add_apply
theorem iteratedFDeriv_add_apply {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 i f) (hg : ContDiff 𝕜 i g) :
    iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 (↑i) f) (hg : ContDiff 𝕜 (↑i) g)
Goal: iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x
-- iteratedFDeriv_add_apply'
theorem iteratedFDeriv_add_apply' {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 i f)
    (hg : ContDiff 𝕜 i g) :
    iteratedFDeriv 𝕜 i (fun x => f x + g x) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {i : ℕ} {f g : E → F} (hf : ContDiff 𝕜 (↑i) f) (hg : ContDiff 𝕜 (↑i) g)
Goal: iteratedFDeriv 𝕜 i (fun x => f x + g x) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x
-- contDiff_neg
theorem contDiff_neg : ContDiff 𝕜 n fun p : F => -p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞}
Goal: ContDiff 𝕜 n fun p => -p
-- ContDiffWithinAt.neg
theorem ContDiffWithinAt.neg {s : Set E} {f : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n (fun x => -f x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {s : Set E} {f : E → F} (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => -f x) s x
-- ContDiffAt.neg
theorem ContDiffAt.neg {f : E → F} (hf : ContDiffAt 𝕜 n f x) :
    ContDiffAt 𝕜 n (fun x => -f x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {f : E → F} (hf : ContDiffAt 𝕜 n f x)
Goal: ContDiffAt 𝕜 n (fun x => -f x) x
-- ContDiff.neg
theorem ContDiff.neg {f : E → F} (hf : ContDiff 𝕜 n f) : ContDiff 𝕜 n fun x => -f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → F} (hf : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n fun x => -f x
-- ContDiffOn.neg
theorem ContDiffOn.neg {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s) :
    ContDiffOn 𝕜 n (fun x => -f x) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {s : Set E} {f : E → F} (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (fun x => -f x) s
-- iteratedFDerivWithin_neg_apply
theorem iteratedFDerivWithin_neg_apply {f : E → F} (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {x : E} {i : ℕ} {f : E → F} (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x
-- iteratedFDeriv_neg_apply
theorem iteratedFDeriv_neg_apply {i : ℕ} {f : E → F} :
    iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {i : ℕ} {f : E → F}
Goal: iteratedFDeriv 𝕜 i (-f) x = -iteratedFDeriv 𝕜 i f x
-- ContDiffWithinAt.sub
theorem ContDiffWithinAt.sub {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x - g x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {s : Set E} {f g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => f x - g x) s x
-- ContDiffAt.sub
theorem ContDiffAt.sub {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) :
    ContDiffAt 𝕜 n (fun x => f x - g x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {f g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x)
Goal: ContDiffAt 𝕜 n (fun x => f x - g x) x
-- ContDiffOn.sub
theorem ContDiffOn.sub {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s)
    (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x - g x) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {s : Set E} {f g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s)
Goal: ContDiffOn 𝕜 n (fun x => f x - g x) s
-- ContDiff.sub
theorem ContDiff.sub {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n fun x => f x - g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n fun x => f x - g x
-- ContDiffWithinAt.sum
theorem ContDiffWithinAt.sum {ι : Type*} {f : ι → E → F} {s : Finset ι} {t : Set E} {x : E}
    (h : ∀ i ∈ s, ContDiffWithinAt 𝕜 n (fun x => f i x) t x) :
    ContDiffWithinAt 𝕜 n (fun x => ∑ i ∈ s, f i x) t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {ι : Type u_3} {f : ι → E → F} {s : Finset ι} {t : Set E} {x : E} (h : ∀ i ∈ s, ContDiffWithinAt 𝕜 n (fun x => f i x) t x)
Goal: ContDiffWithinAt 𝕜 n (fun x => ∑ i ∈ s, f i x) t x
-- ContDiffAt.sum
theorem ContDiffAt.sum {ι : Type*} {f : ι → E → F} {s : Finset ι} {x : E}
    (h : ∀ i ∈ s, ContDiffAt 𝕜 n (fun x => f i x) x) :
    ContDiffAt 𝕜 n (fun x => ∑ i ∈ s, f i x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {ι : Type u_3} {f : ι → E → F} {s : Finset ι} {x : E} (h : ∀ i ∈ s, ContDiffAt 𝕜 n (fun x => f i x) x)
Goal: ContDiffAt 𝕜 n (fun x => ∑ i ∈ s, f i x) x
-- ContDiffOn.sum
theorem ContDiffOn.sum {ι : Type*} {f : ι → E → F} {s : Finset ι} {t : Set E}
    (h : ∀ i ∈ s, ContDiffOn 𝕜 n (fun x => f i x) t) :
    ContDiffOn 𝕜 n (fun x => ∑ i ∈ s, f i x) t 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {ι : Type u_3} {f : ι → E → F} {s : Finset ι} {t : Set E} (h : ∀ i ∈ s, ContDiffOn 𝕜 n (fun x => f i x) t)
Goal: ContDiffOn 𝕜 n (fun x => ∑ i ∈ s, f i x) t
-- ContDiff.sum
theorem ContDiff.sum {ι : Type*} {f : ι → E → F} {s : Finset ι}
    (h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x) : ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {ι : Type u_3} {f : ι → E → F} {s : Finset ι} (h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x)
Goal: ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x
-- iteratedFDerivWithin_sum_apply
theorem iteratedFDerivWithin_sum_apply {ι : Type*} {f : ι → E → F} {u : Finset ι} {i : ℕ} {x : E}
    (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : ∀ j ∈ u, ContDiffOn 𝕜 i (f j) s) :
    iteratedFDerivWithin 𝕜 i (∑ j ∈ u, f j ·) s x =
      ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {ι : Type u_3} {f : ι → E → F} {u : Finset ι} {i : ℕ} {x : E} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) (h : ∀ j ∈ u, ContDiffOn 𝕜 (↑i) (f j) s)
Goal: iteratedFDerivWithin 𝕜 i (fun x => ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x
-- iteratedFDeriv_sum
theorem iteratedFDeriv_sum {ι : Type*} {f : ι → E → F} {u : Finset ι} {i : ℕ}
    (h : ∀ j ∈ u, ContDiff 𝕜 i (f j)) :
    iteratedFDeriv 𝕜 i (∑ j ∈ u, f j ·) = ∑ j ∈ u, iteratedFDeriv 𝕜 i (f j) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_3} {f : ι → E → F} {u : Finset ι} {i : ℕ} (h : ∀ j ∈ u, ContDiff 𝕜 (↑i) (f j))
Goal: (iteratedFDeriv 𝕜 i fun x => ∑ j ∈ u, f j x) = ∑ j ∈ u, iteratedFDeriv 𝕜 i (f j)
-- contDiff_mul
theorem contDiff_mul : ContDiff 𝕜 n fun p : 𝔸 × 𝔸 => p.1 * p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸]
Goal: ContDiff 𝕜 n fun p => p.1 * p.2