ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasDerivAt_zpow
theorem hasDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    HasDerivAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m)
Goal: HasDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x
-- hasDerivWithinAt_zpow
theorem hasDerivWithinAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) (s : Set 𝕜) :
    HasDerivWithinAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) (s : Set 𝕜)
Goal: HasDerivWithinAt (fun x => x ^ m) (↑m * x ^ (m - 1)) s x
-- differentiableAt_zpow
theorem differentiableAt_zpow : DifferentiableAt 𝕜 (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {m : ℤ}
Goal: DifferentiableAt 𝕜 (fun x => x ^ m) x ↔ x ≠ 0 ∨ 0 ≤ m
-- differentiableWithinAt_zpow
theorem differentiableWithinAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    DifferentiableWithinAt 𝕜 (fun x => x ^ m) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m)
Goal: DifferentiableWithinAt 𝕜 (fun x => x ^ m) s x
-- differentiableOn_zpow
theorem differentiableOn_zpow (m : ℤ) (s : Set 𝕜) (h : (0 : 𝕜) ∉ s ∨ 0 ≤ m) :
    DifferentiableOn 𝕜 (fun x => x ^ m) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (s : Set 𝕜) (h : 0 ∉ s ∨ 0 ≤ m)
Goal: DifferentiableOn 𝕜 (fun x => x ^ m) s
-- deriv_zpow
theorem deriv_zpow (m : ℤ) (x : 𝕜) : deriv (fun x => x ^ m) x = m * x ^ (m - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (x : 𝕜)
Goal: deriv (fun x => x ^ m) x = ↑m * x ^ (m - 1)
-- deriv_zpow'
@[simp]
theorem deriv_zpow' (m : ℤ) : (deriv fun x : 𝕜 => x ^ m) = fun x => (m : 𝕜) * x ^ (m - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ)
Goal: (deriv fun x => x ^ m) = fun x => ↑m * x ^ (m - 1)
-- derivWithin_zpow
theorem derivWithin_zpow (hxs : UniqueDiffWithinAt 𝕜 s x) (h : x ≠ 0 ∨ 0 ≤ m) :
    derivWithin (fun x => x ^ m) s x = (m : 𝕜) * x ^ (m - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {m : ℤ} (hxs : UniqueDiffWithinAt 𝕜 s x) (h : x ≠ 0 ∨ 0 ≤ m)
Goal: derivWithin (fun x => x ^ m) s x = ↑m * x ^ (m - 1)
-- iter_deriv_zpow'
@[simp]
theorem iter_deriv_zpow' (m : ℤ) (k : ℕ) :
    (deriv^[k] fun x : 𝕜 => x ^ m) =
      fun x => (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (k : ℕ)
Goal: (deriv^[k] fun x => x ^ m) = fun x => (∏ i ∈ Finset.range k, (↑m - ↑i)) * x ^ (m - ↑k)
-- iter_deriv_zpow
theorem iter_deriv_zpow (m : ℤ) (x : 𝕜) (k : ℕ) :
    deriv^[k] (fun y => y ^ m) x = (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (x : 𝕜) (k : ℕ)
Goal: deriv^[k] (fun y => y ^ m) x = (∏ i ∈ Finset.range k, (↑m - ↑i)) * x ^ (m - ↑k)
-- iter_deriv_pow
theorem iter_deriv_pow (n : ℕ) (x : 𝕜) (k : ℕ) :
    deriv^[k] (fun x : 𝕜 => x ^ n) x = (∏ i ∈ Finset.range k, ((n : 𝕜) - i)) * x ^ (n - k) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ) (x : 𝕜) (k : ℕ)
Goal: deriv^[k] (fun x => x ^ n) x = (∏ i ∈ Finset.range k, (↑n - ↑i)) * x ^ (n - k)
-- iter_deriv_pow'
@[simp]
theorem iter_deriv_pow' (n k : ℕ) :
    (deriv^[k] fun x : 𝕜 => x ^ n) =
      fun x => (∏ i ∈ Finset.range k, ((n : 𝕜) - i)) * x ^ (n - k) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n k : ℕ)
Goal: (deriv^[k] fun x => x ^ n) = fun x => (∏ i ∈ Finset.range k, (↑n - ↑i)) * x ^ (n - k)
-- iter_deriv_inv
theorem iter_deriv_inv (k : ℕ) (x : 𝕜) :
    deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (k : ℕ) (x : 𝕜)
Goal: deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)
-- iter_deriv_inv'
@[simp]
theorem iter_deriv_inv' (k : ℕ) :
    deriv^[k] Inv.inv = fun x : 𝕜 => (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (k : ℕ)
Goal: deriv^[k] Inv.inv = fun x => (∏ i ∈ Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)
-- DifferentiableWithinAt.zpow
theorem DifferentiableWithinAt.zpow (hf : DifferentiableWithinAt 𝕜 f t a) (h : f a ≠ 0 ∨ 0 ≤ m) :
    DifferentiableWithinAt 𝕜 (fun x => f x ^ m) t a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {m : ℤ} {f : E → 𝕜} {t : Set E} {a : E} (hf : DifferentiableWithinAt 𝕜 f t a) (h : f a ≠ 0 ∨ 0 ≤ m)
Goal: DifferentiableWithinAt 𝕜 (fun x => f x ^ m) t a
-- DifferentiableAt.zpow
theorem DifferentiableAt.zpow (hf : DifferentiableAt 𝕜 f a) (h : f a ≠ 0 ∨ 0 ≤ m) :
    DifferentiableAt 𝕜 (fun x => f x ^ m) a 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {m : ℤ} {f : E → 𝕜} {a : E} (hf : DifferentiableAt 𝕜 f a) (h : f a ≠ 0 ∨ 0 ≤ m)
Goal: DifferentiableAt 𝕜 (fun x => f x ^ m) a
-- DifferentiableOn.zpow
theorem DifferentiableOn.zpow (hf : DifferentiableOn 𝕜 f t) (h : (∀ x ∈ t, f x ≠ 0) ∨ 0 ≤ m) :
    DifferentiableOn 𝕜 (fun x => f x ^ m) t 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {m : ℤ} {f : E → 𝕜} {t : Set E} (hf : DifferentiableOn 𝕜 f t) (h : (∀ x ∈ t, f x ≠ 0) ∨ 0 ≤ m)
Goal: DifferentiableOn 𝕜 (fun x => f x ^ m) t
-- Differentiable.zpow
theorem Differentiable.zpow (hf : Differentiable 𝕜 f) (h : (∀ x, f x ≠ 0) ∨ 0 ≤ m) :
    Differentiable 𝕜 fun x => f x ^ m 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {m : ℤ} {f : E → 𝕜} (hf : Differentiable 𝕜 f) (h : (∀ (x : E), f x ≠ 0) ∨ 0 ≤ m)
Goal: Differentiable 𝕜 fun x => f x ^ m
-- HasDerivAtFilter.prod
nonrec theorem HasDerivAtFilter.prod (hf₁ : HasDerivAtFilter f₁ f₁' x L)
    (hf₂ : HasDerivAtFilter f₂ f₂' x L) : HasDerivAtFilter (fun x => (f₁ x, f₂ x)) (f₁', f₂') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₁ : 𝕜 → F} {f₁' : F} {x : 𝕜} {L : Filter 𝕜} {G : Type w} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : 𝕜 → G} {f₂' : G} (hf₁ : HasDerivAtFilter f₁ f₁' x L) (hf₂ : HasDerivAtFilter f₂ f₂' x L)
Goal: HasDerivAtFilter (fun x => (f₁ x, f₂ x)) (f₁', f₂') x L
-- HasDerivWithinAt.prod
nonrec theorem HasDerivWithinAt.prod (hf₁ : HasDerivWithinAt f₁ f₁' s x)
    (hf₂ : HasDerivWithinAt f₂ f₂' s x) : HasDerivWithinAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₁ : 𝕜 → F} {f₁' : F} {x : 𝕜} {s : Set 𝕜} {G : Type w} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : 𝕜 → G} {f₂' : G} (hf₁ : HasDerivWithinAt f₁ f₁' s x) (hf₂ : HasDerivWithinAt f₂ f₂' s x)
Goal: HasDerivWithinAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') s x
-- HasDerivAt.prod
nonrec theorem HasDerivAt.prod (hf₁ : HasDerivAt f₁ f₁' x) (hf₂ : HasDerivAt f₂ f₂' x) :
    HasDerivAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₁ : 𝕜 → F} {f₁' : F} {x : 𝕜} {G : Type w} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : 𝕜 → G} {f₂' : G} (hf₁ : HasDerivAt f₁ f₁' x) (hf₂ : HasDerivAt f₂ f₂' x)
Goal: HasDerivAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') x
-- HasStrictDerivAt.prod
nonrec theorem HasStrictDerivAt.prod (hf₁ : HasStrictDerivAt f₁ f₁' x)
    (hf₂ : HasStrictDerivAt f₂ f₂' x) : HasStrictDerivAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₁ : 𝕜 → F} {f₁' : F} {x : 𝕜} {G : Type w} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : 𝕜 → G} {f₂' : G} (hf₁ : HasStrictDerivAt f₁ f₁' x) (hf₂ : HasStrictDerivAt f₂ f₂' x)
Goal: HasStrictDerivAt (fun x => (f₁ x, f₂ x)) (f₁', f₂') x
-- hasStrictDerivAt_pi
@[simp]
theorem hasStrictDerivAt_pi :
    HasStrictDerivAt φ φ' x ↔ ∀ i, HasStrictDerivAt (fun x => φ x i) (φ' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} {φ' : (i : ι) → E' i}
Goal: HasStrictDerivAt φ φ' x ↔ ∀ (i : ι), HasStrictDerivAt (fun x => φ x i) (φ' i) x
-- hasDerivAtFilter_pi
@[simp]
theorem hasDerivAtFilter_pi :
    HasDerivAtFilter φ φ' x L ↔ ∀ i, HasDerivAtFilter (fun x => φ x i) (φ' i) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {L : Filter 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} {φ' : (i : ι) → E' i}
Goal: HasDerivAtFilter φ φ' x L ↔ ∀ (i : ι), HasDerivAtFilter (fun x => φ x i) (φ' i) x L
-- hasDerivAt_pi
theorem hasDerivAt_pi : HasDerivAt φ φ' x ↔ ∀ i, HasDerivAt (fun x => φ x i) (φ' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} {φ' : (i : ι) → E' i}
Goal: HasDerivAt φ φ' x ↔ ∀ (i : ι), HasDerivAt (fun x => φ x i) (φ' i) x
-- hasDerivWithinAt_pi
theorem hasDerivWithinAt_pi :
    HasDerivWithinAt φ φ' s x ↔ ∀ i, HasDerivWithinAt (fun x => φ x i) (φ' i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} {φ' : (i : ι) → E' i}
Goal: HasDerivWithinAt φ φ' s x ↔ ∀ (i : ι), HasDerivWithinAt (fun x => φ x i) (φ' i) s x
-- derivWithin_pi
theorem derivWithin_pi (h : ∀ i, DifferentiableWithinAt 𝕜 (fun x => φ x i) s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin φ s x = fun i => derivWithin (fun x => φ x i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} (h : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => φ x i) s x) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin φ s x = fun i => derivWithin (fun x => φ x i) s x
-- deriv_pi
theorem deriv_pi (h : ∀ i, DifferentiableAt 𝕜 (fun x => φ x i) x) :
    deriv φ x = fun i => deriv (fun x => φ x i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_1} [Fintype ι] {E' : ι → Type u_2} [(i : ι) → NormedAddCommGroup (E' i)] [(i : ι) → NormedSpace 𝕜 (E' i)] {φ : 𝕜 → (i : ι) → E' i} (h : ∀ (i : ι), DifferentiableAt 𝕜 (fun x => φ x i) x)
Goal: deriv φ x = fun i => deriv (fun x => φ x i) x
-- ContinuousLinearMap.hasDerivAtFilter
protected theorem ContinuousLinearMap.hasDerivAtFilter : HasDerivAtFilter e (e 1) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {L : Filter 𝕜} (e : 𝕜 →L[𝕜] F)
Goal: HasDerivAtFilter (⇑e) (e 1) x L
-- ContinuousLinearMap.hasStrictDerivAt
protected theorem ContinuousLinearMap.hasStrictDerivAt : HasStrictDerivAt e (e 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →L[𝕜] F)
Goal: HasStrictDerivAt (⇑e) (e 1) x
-- ContinuousLinearMap.hasDerivAt
protected theorem ContinuousLinearMap.hasDerivAt : HasDerivAt e (e 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →L[𝕜] F)
Goal: HasDerivAt (⇑e) (e 1) x
-- ContinuousLinearMap.hasDerivWithinAt
protected theorem ContinuousLinearMap.hasDerivWithinAt : HasDerivWithinAt e (e 1) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} (e : 𝕜 →L[𝕜] F)
Goal: HasDerivWithinAt (⇑e) (e 1) s x
-- ContinuousLinearMap.deriv
@[simp]
protected theorem ContinuousLinearMap.deriv : deriv e x = e 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →L[𝕜] F)
Goal: deriv (⇑e) x = e 1
-- ContinuousLinearMap.derivWithin
protected theorem ContinuousLinearMap.derivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin e s x = e 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} (e : 𝕜 →L[𝕜] F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (⇑e) s x = e 1
-- LinearMap.hasDerivAtFilter
protected theorem LinearMap.hasDerivAtFilter : HasDerivAtFilter e (e 1) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {L : Filter 𝕜} (e : 𝕜 →ₗ[𝕜] F)
Goal: HasDerivAtFilter (⇑e) (e 1) x L
-- LinearMap.hasStrictDerivAt
protected theorem LinearMap.hasStrictDerivAt : HasStrictDerivAt e (e 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →ₗ[𝕜] F)
Goal: HasStrictDerivAt (⇑e) (e 1) x
-- LinearMap.hasDerivAt
protected theorem LinearMap.hasDerivAt : HasDerivAt e (e 1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →ₗ[𝕜] F)
Goal: HasDerivAt (⇑e) (e 1) x
-- LinearMap.hasDerivWithinAt
protected theorem LinearMap.hasDerivWithinAt : HasDerivWithinAt e (e 1) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} (e : 𝕜 →ₗ[𝕜] F)
Goal: HasDerivWithinAt (⇑e) (e 1) s x
-- LinearMap.deriv
@[simp]
protected theorem LinearMap.deriv : deriv e x = e 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} (e : 𝕜 →ₗ[𝕜] F)
Goal: deriv (⇑e) x = e 1
-- LinearMap.derivWithin
protected theorem LinearMap.derivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin e s x = e 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} (e : 𝕜 →ₗ[𝕜] F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (⇑e) s x = e 1
-- hasStrictDerivAt_pow
theorem hasStrictDerivAt_pow :
    ∀ (n : ℕ) (x : 𝕜), HasStrictDerivAt (fun x : 𝕜 ↦ x ^ n) ((n : 𝕜) * x ^ (n - 1)) x
  
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ) (x : 𝕜)
Goal: HasStrictDerivAt (fun x => x ^ n) (↑n * x ^ (n - 1)) x
-- hasDerivAt_pow
theorem hasDerivAt_pow (n : ℕ) (x : 𝕜) :
    HasDerivAt (fun x : 𝕜 => x ^ n) ((n : 𝕜) * x ^ (n - 1)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ) (x : 𝕜)
Goal: HasDerivAt (fun x => x ^ n) (↑n * x ^ (n - 1)) x
-- hasDerivWithinAt_pow
theorem hasDerivWithinAt_pow (n : ℕ) (x : 𝕜) (s : Set 𝕜) :
    HasDerivWithinAt (fun x : 𝕜 => x ^ n) ((n : 𝕜) * x ^ (n - 1)) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ) (x : 𝕜) (s : Set 𝕜)
Goal: HasDerivWithinAt (fun x => x ^ n) (↑n * x ^ (n - 1)) s x
-- differentiableAt_pow
theorem differentiableAt_pow : DifferentiableAt 𝕜 (fun x : 𝕜 => x ^ n) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (n : ℕ)
Goal: DifferentiableAt 𝕜 (fun x => x ^ n) x
-- differentiableWithinAt_pow
theorem differentiableWithinAt_pow :
    DifferentiableWithinAt 𝕜 (fun x : 𝕜 => x ^ n) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (n : ℕ)
Goal: DifferentiableWithinAt 𝕜 (fun x => x ^ n) s x
-- differentiable_pow
theorem differentiable_pow : Differentiable 𝕜 fun x : 𝕜 => x ^ n 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ)
Goal: Differentiable 𝕜 fun x => x ^ n
-- differentiableOn_pow
theorem differentiableOn_pow : DifferentiableOn 𝕜 (fun x : 𝕜 => x ^ n) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} (n : ℕ)
Goal: DifferentiableOn 𝕜 (fun x => x ^ n) s
-- deriv_pow
theorem deriv_pow : deriv (fun x : 𝕜 => x ^ n) x = (n : 𝕜) * x ^ (n - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (n : ℕ)
Goal: deriv (fun x => x ^ n) x = ↑n * x ^ (n - 1)
-- deriv_pow'
@[simp]
theorem deriv_pow' : (deriv fun x : 𝕜 => x ^ n) = fun x => (n : 𝕜) * x ^ (n - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (n : ℕ)
Goal: (deriv fun x => x ^ n) = fun x => ↑n * x ^ (n - 1)
-- derivWithin_pow
theorem derivWithin_pow (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x : 𝕜 => x ^ n) s x = (n : 𝕜) * x ^ (n - 1) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} (n : ℕ) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => x ^ n) s x = ↑n * x ^ (n - 1)
-- HasDerivWithinAt.pow
theorem HasDerivWithinAt.pow (hc : HasDerivWithinAt c c' s x) :
    HasDerivWithinAt (fun y => c y ^ n) ((n : 𝕜) * c x ^ (n - 1) * c') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {c : 𝕜 → 𝕜} {c' : 𝕜} (n : ℕ) (hc : HasDerivWithinAt c c' s x)
Goal: HasDerivWithinAt (fun y => c y ^ n) (↑n * c x ^ (n - 1) * c') s x
-- HasDerivAt.pow
theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :
    HasDerivAt (fun y => c y ^ n) ((n : 𝕜) * c x ^ (n - 1) * c') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {c : 𝕜 → 𝕜} {c' : 𝕜} (n : ℕ) (hc : HasDerivAt c c' x)
Goal: HasDerivAt (fun y => c y ^ n) (↑n * c x ^ (n - 1) * c') x
-- derivWithin_pow'
theorem derivWithin_pow' (hc : DifferentiableWithinAt 𝕜 c s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => c x ^ n) s x = (n : 𝕜) * c x ^ (n - 1) * derivWithin c s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {c : 𝕜 → 𝕜} (n : ℕ) (hc : DifferentiableWithinAt 𝕜 c s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => c x ^ n) s x = ↑n * c x ^ (n - 1) * derivWithin c s x
-- deriv_pow''
@[simp]
theorem deriv_pow'' (hc : DifferentiableAt 𝕜 c x) :
    deriv (fun x => c x ^ n) x = (n : 𝕜) * c x ^ (n - 1) * deriv c x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {c : 𝕜 → 𝕜} (n : ℕ) (hc : DifferentiableAt 𝕜 c x)
Goal: deriv (fun x => c x ^ n) x = ↑n * c x ^ (n - 1) * deriv c x
-- Unitization.splitMul_apply
@[simp]
theorem splitMul_apply (x : Unitization 𝕜 A) :
    splitMul 𝕜 A x = (x.fst, algebraMap 𝕜 (A →L[𝕜] A) x.fst + mul 𝕜 A x.snd) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (x : Unitization 𝕜 A)
Goal: (Unitization.splitMul 𝕜 A) x = (x.fst, (algebraMap 𝕜 (A →L[𝕜] A)) x.fst + (ContinuousLinearMap.mul 𝕜 A) x.snd)
-- Unitization.splitMul_injective_of_clm_mul_injective
theorem splitMul_injective_of_clm_mul_injective
    (h : Function.Injective (mul 𝕜 A)) :
    Function.Injective (splitMul 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] (h : Function.Injective ⇑(ContinuousLinearMap.mul 𝕜 A))
Goal: Function.Injective ⇑(Unitization.splitMul 𝕜 A)
-- Unitization.splitMul_injective
theorem splitMul_injective : Function.Injective (splitMul 𝕜 A) 
Arguments: (𝕜 : Type u_1) (A : Type u_2) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: Function.Injective ⇑(Unitization.splitMul 𝕜 A)
-- Unitization.norm_def
theorem norm_def (x : Unitization 𝕜 A) : ‖x‖ = ‖splitMul 𝕜 A x‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (x : Unitization 𝕜 A)
Goal: ‖x‖ = ‖(Unitization.splitMul 𝕜 A) x‖
-- Unitization.nnnorm_def
theorem nnnorm_def (x : Unitization 𝕜 A) : ‖x‖₊ = ‖splitMul 𝕜 A x‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (x : Unitization 𝕜 A)
Goal: ‖x‖₊ = ‖(Unitization.splitMul 𝕜 A) x‖₊
-- Unitization.norm_eq_sup
theorem norm_eq_sup (x : Unitization 𝕜 A) :
    ‖x‖ = ‖x.fst‖ ⊔ ‖algebraMap 𝕜 (A →L[𝕜] A) x.fst + mul 𝕜 A x.snd‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (x : Unitization 𝕜 A)
Goal: ‖x‖ = ‖x.fst‖ ⊔ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.fst + (ContinuousLinearMap.mul 𝕜 A) x.snd‖
-- Unitization.nnnorm_eq_sup
theorem nnnorm_eq_sup (x : Unitization 𝕜 A) :
    ‖x‖₊ = ‖x.fst‖₊ ⊔ ‖algebraMap 𝕜 (A →L[𝕜] A) x.fst + mul 𝕜 A x.snd‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (x : Unitization 𝕜 A)
Goal: ‖x‖₊ = ‖x.fst‖₊ ⊔ ‖(algebraMap 𝕜 (A →L[𝕜] A)) x.fst + (ContinuousLinearMap.mul 𝕜 A) x.snd‖₊
-- Unitization.lipschitzWith_addEquiv
theorem lipschitzWith_addEquiv :
    LipschitzWith 2 (Unitization.addEquiv 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: LipschitzWith 2 ⇑(Unitization.addEquiv 𝕜 A)
-- Unitization.antilipschitzWith_addEquiv
theorem antilipschitzWith_addEquiv :
    AntilipschitzWith 2 (addEquiv 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: AntilipschitzWith 2 ⇑(Unitization.addEquiv 𝕜 A)
-- Unitization.uniformity_eq_aux
theorem uniformity_eq_aux :
    𝓤[instUniformSpaceProd.comap <| addEquiv 𝕜 A] = 𝓤 (Unitization 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: uniformity (Unitization 𝕜 A) = uniformity (Unitization 𝕜 A)
-- Unitization.cobounded_eq_aux
theorem cobounded_eq_aux :
    @cobounded _ (Bornology.induced <| addEquiv 𝕜 A) = cobounded (Unitization 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: Bornology.cobounded (Unitization 𝕜 A) = Bornology.cobounded (Unitization 𝕜 A)
-- Unitization.uniformEmbedding_addEquiv
theorem uniformEmbedding_addEquiv : UniformEmbedding (addEquiv 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
Goal: UniformEmbedding ⇑(Unitization.addEquiv 𝕜 A)
-- Unitization.norm_inr
lemma norm_inr (a : A) : ‖(a : Unitization 𝕜 A)‖ = ‖a‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (a : A)
Goal: ‖↑a‖ = ‖a‖
-- Unitization.nnnorm_inr
lemma nnnorm_inr (a : A) : ‖(a : Unitization 𝕜 A)‖₊ = ‖a‖₊ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (a : A)
Goal: ‖↑a‖₊ = ‖a‖₊
-- Unitization.isometry_inr
lemma isometry_inr : Isometry ((↑) : A → Unitization 𝕜 A) 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A]
Goal: Isometry Unitization.inr
-- Unitization.dist_inr
lemma dist_inr (a b : A) : dist (a : Unitization 𝕜 A) (b : Unitization 𝕜 A) = dist a b 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (a b : A)
Goal: dist ↑a ↑b = dist a b
-- Unitization.nndist_inr
lemma nndist_inr (a b : A) : nndist (a : Unitization 𝕜 A) (b : Unitization 𝕜 A) = nndist a b 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A] [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [RegularNormedAlgebra 𝕜 A] (a b : A)
Goal: nndist ↑a ↑b = nndist a b
-- hasStrictDerivAt_inv
theorem hasStrictDerivAt_inv (hx : x ≠ 0) : HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (hx : x ≠ 0)
Goal: HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x