ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasStrictFDerivAt_finset_prod
theorem hasStrictFDerivAt_finset_prod [DecidableEq ι] [Fintype ι] {x : ι → 𝔸'} :
    HasStrictFDerivAt (𝕜 := 𝕜) (∏ i ∈ u, · i) (∑ i ∈ u, (∏ j ∈ u.erase i, x j) • proj i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} [DecidableEq ι] [Fintype ι] {x : ι → 𝔸'}
Goal: HasStrictFDerivAt (fun x => ∏ i ∈ u, x i) (∑ i ∈ u, (∏ j ∈ u.erase i, x j) • ContinuousLinearMap.proj i) x
-- hasFDerivAt_finset_prod
theorem hasFDerivAt_finset_prod [DecidableEq ι] [Fintype ι] {x : ι → 𝔸'} :
    HasFDerivAt (𝕜 := 𝕜) (∏ i ∈ u, · i) (∑ i ∈ u, (∏ j ∈ u.erase i, x j) • proj i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} [DecidableEq ι] [Fintype ι] {x : ι → 𝔸'}
Goal: HasFDerivAt (fun x => ∏ i ∈ u, x i) (∑ i ∈ u, (∏ j ∈ u.erase i, x j) • ContinuousLinearMap.proj i) x
-- HasStrictFDerivAt.list_prod'
@[fun_prop]
theorem HasStrictFDerivAt.list_prod' {l : List ι} {x : E}
    (h : ∀ i ∈ l, HasStrictFDerivAt (f i ·) (f' i) x) :
    HasStrictFDerivAt (fun x ↦ (l.map (f · x)).prod)
      (∑ i : Fin l.length, ((l.take i).map (f · x)).prod •
        smulRight (f' (l.get i)) ((l.drop (.succ i)).map (f · x)).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : ι → E → 𝔸} {f' : ι → E →L[𝕜] 𝔸} {l : List ι} {x : E} (h : ∀ i ∈ l, HasStrictFDerivAt (fun x => f i x) (f' i) x)
Goal: HasStrictFDerivAt (fun x => (List.map (fun x_1 => f x_1 x) l).prod)
  (∑ i : Fin l.length,
    (List.map (fun x_1 => f x_1 x) (List.take (↑i) l)).prod •
      (f' (l.get i)).smulRight (List.map (fun x_1 => f x_1 x) (List.drop (↑i).succ l)).prod)
  x
-- HasFDerivAt.list_prod'
@[fun_prop]
theorem HasFDerivAt.list_prod' {l : List ι} {x : E}
    (h : ∀ i ∈ l, HasFDerivAt (f i ·) (f' i) x) :
    HasFDerivAt (fun x ↦ (l.map (f · x)).prod)
      (∑ i : Fin l.length, ((l.take i).map (f · x)).prod •
        smulRight (f' (l.get i)) ((l.drop (.succ i)).map (f · x)).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : ι → E → 𝔸} {f' : ι → E →L[𝕜] 𝔸} {l : List ι} {x : E} (h : ∀ i ∈ l, HasFDerivAt (fun x => f i x) (f' i) x)
Goal: HasFDerivAt (fun x => (List.map (fun x_1 => f x_1 x) l).prod)
  (∑ i : Fin l.length,
    (List.map (fun x_1 => f x_1 x) (List.take (↑i) l)).prod •
      (f' (l.get i)).smulRight (List.map (fun x_1 => f x_1 x) (List.drop (↑i).succ l)).prod)
  x
-- HasFDerivWithinAt.list_prod'
@[fun_prop]
theorem HasFDerivWithinAt.list_prod' {l : List ι} {x : E}
    (h : ∀ i ∈ l, HasFDerivWithinAt (f i ·) (f' i) s x) :
    HasFDerivWithinAt (fun x ↦ (l.map (f · x)).prod)
      (∑ i : Fin l.length, ((l.take i).map (f · x)).prod •
        smulRight (f' (l.get i)) ((l.drop (.succ i)).map (f · x)).prod) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : ι → E → 𝔸} {f' : ι → E →L[𝕜] 𝔸} {l : List ι} {x : E} (h : ∀ i ∈ l, HasFDerivWithinAt (fun x => f i x) (f' i) s x)
Goal: HasFDerivWithinAt (fun x => (List.map (fun x_1 => f x_1 x) l).prod)
  (∑ i : Fin l.length,
    (List.map (fun x_1 => f x_1 x) (List.take (↑i) l)).prod •
      (f' (l.get i)).smulRight (List.map (fun x_1 => f x_1 x) (List.drop (↑i).succ l)).prod)
  s x
-- fderiv_list_prod'
theorem fderiv_list_prod' {l : List ι} {x : E}
    (h : ∀ i ∈ l, DifferentiableAt 𝕜 (f i ·) x) :
    fderiv 𝕜 (fun x ↦ (l.map (f · x)).prod) x =
      ∑ i : Fin l.length, ((l.take i).map (f · x)).prod •
        smulRight (fderiv 𝕜 (fun x ↦ f (l.get i) x) x)
          ((l.drop (.succ i)).map (f · x)).prod 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : ι → E → 𝔸} {l : List ι} {x : E} (h : ∀ i ∈ l, DifferentiableAt 𝕜 (fun x => f i x) x)
Goal: fderiv 𝕜 (fun x => (List.map (fun x_1 => f x_1 x) l).prod) x =
  ∑ i : Fin l.length,
    (List.map (fun x_1 => f x_1 x) (List.take (↑i) l)).prod •
      (fderiv 𝕜 (fun x => f (l.get i) x) x).smulRight (List.map (fun x_1 => f x_1 x) (List.drop (↑i).succ l)).prod
-- fderivWithin_list_prod'
theorem fderivWithin_list_prod' {l : List ι} {x : E}
    (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ l, DifferentiableWithinAt 𝕜 (f i ·) s x) :
    fderivWithin 𝕜 (fun x ↦ (l.map (f · x)).prod) s x =
      ∑ i : Fin l.length, ((l.take i).map (f · x)).prod •
        smulRight (fderivWithin 𝕜 (fun x ↦ f (l.get i) x) s x)
          ((l.drop (.succ i)).map (f · x)).prod 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : ι → E → 𝔸} {l : List ι} {x : E} (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ l, DifferentiableWithinAt 𝕜 (fun x => f i x) s x)
Goal: fderivWithin 𝕜 (fun x => (List.map (fun x_1 => f x_1 x) l).prod) s x =
  ∑ i : Fin l.length,
    (List.map (fun x_1 => f x_1 x) (List.take (↑i) l)).prod •
      (fderivWithin 𝕜 (fun x => f (l.get i) x) s x).smulRight
        (List.map (fun x_1 => f x_1 x) (List.drop (↑i).succ l)).prod
-- HasStrictFDerivAt.multiset_prod
@[fun_prop]
theorem HasStrictFDerivAt.multiset_prod [DecidableEq ι] {u : Multiset ι} {x : E}
    (h : ∀ i ∈ u, HasStrictFDerivAt (g i ·) (g' i) x) :
    HasStrictFDerivAt (fun x ↦ (u.map (g · x)).prod)
      (u.map fun i ↦ ((u.erase i).map (g · x)).prod • g' i).sum x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {u : Multiset ι} {x : E} (h : ∀ i ∈ u, HasStrictFDerivAt (fun x => g i x) (g' i) x)
Goal: HasStrictFDerivAt (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod)
  (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • g' i) u).sum x
-- HasFDerivAt.multiset_prod
@[fun_prop]
theorem HasFDerivAt.multiset_prod [DecidableEq ι] {u : Multiset ι} {x : E}
    (h : ∀ i ∈ u, HasFDerivAt (g i ·) (g' i) x) :
    HasFDerivAt (fun x ↦ (u.map (g · x)).prod)
      (u.map fun i ↦ ((u.erase i).map (g · x)).prod • g' i).sum x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {u : Multiset ι} {x : E} (h : ∀ i ∈ u, HasFDerivAt (fun x => g i x) (g' i) x)
Goal: HasFDerivAt (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod)
  (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • g' i) u).sum x
-- HasFDerivWithinAt.multiset_prod
@[fun_prop]
theorem HasFDerivWithinAt.multiset_prod [DecidableEq ι] {u : Multiset ι} {x : E}
    (h : ∀ i ∈ u, HasFDerivWithinAt (g i ·) (g' i) s x) :
    HasFDerivWithinAt (fun x ↦ (u.map (g · x)).prod)
      (u.map fun i ↦ ((u.erase i).map (g · x)).prod • g' i).sum s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {u : Multiset ι} {x : E} (h : ∀ i ∈ u, HasFDerivWithinAt (fun x => g i x) (g' i) s x)
Goal: HasFDerivWithinAt (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod)
  (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • g' i) u).sum s x
-- fderiv_multiset_prod
theorem fderiv_multiset_prod [DecidableEq ι] {u : Multiset ι} {x : E}
    (h : ∀ i ∈ u, DifferentiableAt 𝕜 (g i ·) x) :
    fderiv 𝕜 (fun x ↦ (u.map (g · x)).prod) x =
      (u.map fun i ↦ ((u.erase i).map (g · x)).prod • fderiv 𝕜 (g i) x).sum 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} [DecidableEq ι] {u : Multiset ι} {x : E} (h : ∀ i ∈ u, DifferentiableAt 𝕜 (fun x => g i x) x)
Goal: fderiv 𝕜 (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod) x =
  (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • fderiv 𝕜 (g i) x) u).sum
-- fderivWithin_multiset_prod
theorem fderivWithin_multiset_prod [DecidableEq ι] {u : Multiset ι} {x : E}
    (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (g i ·) s x) :
    fderivWithin 𝕜 (fun x ↦ (u.map (g · x)).prod) s x =
      (u.map fun i ↦ ((u.erase i).map (g · x)).prod • fderivWithin 𝕜 (g i) s x).sum 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {g : ι → E → 𝔸'} [DecidableEq ι] {u : Multiset ι} {x : E} (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (fun x => g i x) s x)
Goal: fderivWithin 𝕜 (fun x => (Multiset.map (fun x_1 => g x_1 x) u).prod) s x =
  (Multiset.map (fun i => (Multiset.map (fun x_1 => g x_1 x) (u.erase i)).prod • fderivWithin 𝕜 (g i) s x) u).sum
-- HasStrictFDerivAt.finset_prod
theorem HasStrictFDerivAt.finset_prod [DecidableEq ι] {x : E}
    (hg : ∀ i ∈ u, HasStrictFDerivAt (g i) (g' i) x) :
    HasStrictFDerivAt (∏ i ∈ u, g i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {x : E} (hg : ∀ i ∈ u, HasStrictFDerivAt (g i) (g' i) x)
Goal: HasStrictFDerivAt (fun x => ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x
-- HasFDerivAt.finset_prod
theorem HasFDerivAt.finset_prod [DecidableEq ι] {x : E}
    (hg : ∀ i ∈ u, HasFDerivAt (g i) (g' i) x) :
    HasFDerivAt (∏ i ∈ u, g i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {x : E} (hg : ∀ i ∈ u, HasFDerivAt (g i) (g' i) x)
Goal: HasFDerivAt (fun x => ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x
-- HasFDerivWithinAt.finset_prod
theorem HasFDerivWithinAt.finset_prod [DecidableEq ι] {x : E}
    (hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x) :
    HasFDerivWithinAt (∏ i ∈ u, g i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [DecidableEq ι] {x : E} (hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x)
Goal: HasFDerivWithinAt (fun x => ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) s x
-- fderiv_finset_prod
theorem fderiv_finset_prod [DecidableEq ι] {x : E} (hg : ∀ i ∈ u, DifferentiableAt 𝕜 (g i) x) :
    fderiv 𝕜 (∏ i ∈ u, g i ·) x = ∑ i ∈ u, (∏ j ∈ u.erase i, (g j x)) • fderiv 𝕜 (g i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} [DecidableEq ι] {x : E} (hg : ∀ i ∈ u, DifferentiableAt 𝕜 (g i) x)
Goal: fderiv 𝕜 (fun x => ∏ i ∈ u, g i x) x = ∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • fderiv 𝕜 (g i) x
-- fderivWithin_finset_prod
theorem fderivWithin_finset_prod [DecidableEq ι] {x : E} (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hg : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (g i) s x) :
    fderivWithin 𝕜 (∏ i ∈ u, g i ·) s x =
      ∑ i ∈ u, (∏ j ∈ u.erase i, (g j x)) • fderivWithin 𝕜 (g i) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} [DecidableEq ι] {x : E} (hxs : UniqueDiffWithinAt 𝕜 s x) (hg : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (g i) s x)
Goal: fderivWithin 𝕜 (fun x => ∏ i ∈ u, g i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • fderivWithin 𝕜 (g i) s x
-- hasFDerivAt_ring_inverse
@[fun_prop]
theorem hasFDerivAt_ring_inverse (x : Rˣ) :
    HasFDerivAt Ring.inverse (-mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] (x : Rˣ)
Goal: HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x
-- differentiableAt_inverse
@[fun_prop]
theorem differentiableAt_inverse {x : R} (hx : IsUnit x) :
    DifferentiableAt 𝕜 (@Ring.inverse R _) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : IsUnit x)
Goal: DifferentiableAt 𝕜 Ring.inverse x
-- differentiableWithinAt_inverse
@[fun_prop]
theorem differentiableWithinAt_inverse {x : R} (hx : IsUnit x) (s : Set R) :
    DifferentiableWithinAt 𝕜 (@Ring.inverse R _) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : IsUnit x) (s : Set R)
Goal: DifferentiableWithinAt 𝕜 Ring.inverse s x
-- differentiableOn_inverse
@[fun_prop]
theorem differentiableOn_inverse : DifferentiableOn 𝕜 (@Ring.inverse R _) {x | IsUnit x} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R]
Goal: DifferentiableOn 𝕜 Ring.inverse {x | IsUnit x}
-- fderiv_inverse
theorem fderiv_inverse (x : Rˣ) : fderiv 𝕜 (@Ring.inverse R _) x = -mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] (x : Rˣ)
Goal: fderiv 𝕜 Ring.inverse ↑x = -((ContinuousLinearMap.mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹
-- DifferentiableWithinAt.inverse
@[fun_prop]
theorem DifferentiableWithinAt.inverse (hf : DifferentiableWithinAt 𝕜 h S z) (hz : IsUnit (h z)) :
    DifferentiableWithinAt 𝕜 (fun x => Ring.inverse (h x)) S z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {z : E} {S : Set E} (hf : DifferentiableWithinAt 𝕜 h S z) (hz : IsUnit (h z))
Goal: DifferentiableWithinAt 𝕜 (fun x => Ring.inverse (h x)) S z
-- DifferentiableAt.inverse
@[simp, fun_prop]
theorem DifferentiableAt.inverse (hf : DifferentiableAt 𝕜 h z) (hz : IsUnit (h z)) :
    DifferentiableAt 𝕜 (fun x => Ring.inverse (h x)) z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {z : E} (hf : DifferentiableAt 𝕜 h z) (hz : IsUnit (h z))
Goal: DifferentiableAt 𝕜 (fun x => Ring.inverse (h x)) z
-- DifferentiableOn.inverse
@[fun_prop]
theorem DifferentiableOn.inverse (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, IsUnit (h x)) :
    DifferentiableOn 𝕜 (fun x => Ring.inverse (h x)) S 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {S : Set E} (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, IsUnit (h x))
Goal: DifferentiableOn 𝕜 (fun x => Ring.inverse (h x)) S
-- Differentiable.inverse
@[simp, fun_prop]
theorem Differentiable.inverse (hf : Differentiable 𝕜 h) (hz : ∀ x, IsUnit (h x)) :
    Differentiable 𝕜 fun x => Ring.inverse (h x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} (hf : Differentiable 𝕜 h) (hz : ∀ (x : E), IsUnit (h x))
Goal: Differentiable 𝕜 fun x => Ring.inverse (h x)
-- hasFDerivAt_inv'
@[fun_prop]
theorem hasFDerivAt_inv' {x : R} (hx : x ≠ 0) :
    HasFDerivAt Inv.inv (-mulLeftRight 𝕜 R x⁻¹ x⁻¹) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : x ≠ 0)
Goal: HasFDerivAt Inv.inv (-((ContinuousLinearMap.mulLeftRight 𝕜 R) x⁻¹) x⁻¹) x
-- differentiableAt_inv'
@[fun_prop]
theorem differentiableAt_inv' {x : R} (hx : x ≠ 0) : DifferentiableAt 𝕜 Inv.inv x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : x ≠ 0)
Goal: DifferentiableAt 𝕜 Inv.inv x
-- differentiableWithinAt_inv'
@[fun_prop]
theorem differentiableWithinAt_inv' {x : R} (hx : x ≠ 0) (s : Set R) :
    DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : x ≠ 0) (s : Set R)
Goal: DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x
-- differentiableOn_inv'
@[fun_prop]
theorem differentiableOn_inv' : DifferentiableOn 𝕜 (fun x : R => x⁻¹) {x | x ≠ 0} 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R]
Goal: DifferentiableOn 𝕜 (fun x => x⁻¹) {x | x ≠ 0}
-- fderiv_inv'
theorem fderiv_inv' {x : R} (hx : x ≠ 0) : fderiv 𝕜 Inv.inv x = -mulLeftRight 𝕜 R x⁻¹ x⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {x : R} (hx : x ≠ 0)
Goal: fderiv 𝕜 Inv.inv x = -((ContinuousLinearMap.mulLeftRight 𝕜 R) x⁻¹) x⁻¹
-- fderivWithin_inv'
theorem fderivWithin_inv' {s : Set R} {x : R} (hx : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x => x⁻¹) s x = -mulLeftRight 𝕜 R x⁻¹ x⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {s : Set R} {x : R} (hx : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => x⁻¹) s x = -((ContinuousLinearMap.mulLeftRight 𝕜 R) x⁻¹) x⁻¹
-- DifferentiableWithinAt.inv'
@[fun_prop]
theorem DifferentiableWithinAt.inv' (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
    DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {z : E} {S : Set E} (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0)
Goal: DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z
-- DifferentiableAt.inv'
@[simp, fun_prop]
theorem DifferentiableAt.inv' (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
    DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {z : E} (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0)
Goal: DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z
-- DifferentiableOn.inv'
@[fun_prop]
theorem DifferentiableOn.inv' (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} {S : Set E} (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0)
Goal: DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S
-- Differentiable.inv'
@[simp, fun_prop]
theorem Differentiable.inv' (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
    Differentiable 𝕜 fun x => (h x)⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {R : Type u_6} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] {h : E → R} (hf : Differentiable 𝕜 h) (hz : ∀ (x : E), h x ≠ 0)
Goal: Differentiable 𝕜 fun x => (h x)⁻¹
-- IsLowerSet.nonMemberSubfamily
theorem IsLowerSet.nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :
    IsLowerSet (𝒜.nonMemberSubfamily a : Set (Finset α)) 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} (h : IsLowerSet ↑𝒜)
Goal: IsLowerSet ↑(Finset.nonMemberSubfamily a 𝒜)
-- IsLowerSet.memberSubfamily
theorem IsLowerSet.memberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :
    IsLowerSet (𝒜.memberSubfamily a : Set (Finset α)) 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} (h : IsLowerSet ↑𝒜)
Goal: IsLowerSet ↑(Finset.memberSubfamily a 𝒜)
-- IsLowerSet.memberSubfamily_subset_nonMemberSubfamily
theorem IsLowerSet.memberSubfamily_subset_nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :
    𝒜.memberSubfamily a ⊆ 𝒜.nonMemberSubfamily a 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} (h : IsLowerSet ↑𝒜)
Goal: Finset.memberSubfamily a 𝒜 ⊆ Finset.nonMemberSubfamily a 𝒜
-- IsLowerSet.le_card_inter_finset'
theorem IsLowerSet.le_card_inter_finset' (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))
    (hℬ : IsLowerSet (ℬ : Set (Finset α))) (h𝒜s : ∀ t ∈ 𝒜, t ⊆ s) (hℬs : ∀ t ∈ ℬ, t ⊆ s) :
    𝒜.card * ℬ.card ≤ 2 ^ s.card * (𝒜 ∩ ℬ).card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} (h𝒜 : IsLowerSet ↑𝒜) (hℬ : IsLowerSet ↑ℬ) (h𝒜s : ∀ t ∈ 𝒜, t ⊆ s) (hℬs : ∀ t ∈ ℬ, t ⊆ s)
Goal: 𝒜.card * ℬ.card ≤ 2 ^ s.card * (𝒜 ∩ ℬ).card
-- IsLowerSet.le_card_inter_finset
theorem IsLowerSet.le_card_inter_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))
    (hℬ : IsLowerSet (ℬ : Set (Finset α))) : 𝒜.card * ℬ.card ≤ 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} [Fintype α] (h𝒜 : IsLowerSet ↑𝒜) (hℬ : IsLowerSet ↑ℬ)
Goal: 𝒜.card * ℬ.card ≤ 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card
-- IsUpperSet.card_inter_le_finset
theorem IsUpperSet.card_inter_le_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))
    (hℬ : IsLowerSet (ℬ : Set (Finset α))) :
    2 ^ Fintype.card α * (𝒜 ∩ ℬ).card ≤ 𝒜.card * ℬ.card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} [Fintype α] (h𝒜 : IsUpperSet ↑𝒜) (hℬ : IsLowerSet ↑ℬ)
Goal: 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card ≤ 𝒜.card * ℬ.card
-- IsLowerSet.card_inter_le_finset
theorem IsLowerSet.card_inter_le_finset (h𝒜 : IsLowerSet (𝒜 : Set (Finset α)))
    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :
    2 ^ Fintype.card α * (𝒜 ∩ ℬ).card ≤ 𝒜.card * ℬ.card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} [Fintype α] (h𝒜 : IsLowerSet ↑𝒜) (hℬ : IsUpperSet ↑ℬ)
Goal: 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card ≤ 𝒜.card * ℬ.card
-- IsUpperSet.le_card_inter_finset
theorem IsUpperSet.le_card_inter_finset (h𝒜 : IsUpperSet (𝒜 : Set (Finset α)))
    (hℬ : IsUpperSet (ℬ : Set (Finset α))) :
    𝒜.card * ℬ.card ≤ 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} [Fintype α] (h𝒜 : IsUpperSet ↑𝒜) (hℬ : IsUpperSet ↑ℬ)
Goal: 𝒜.card * ℬ.card ≤ 2 ^ Fintype.card α * (𝒜 ∩ ℬ).card
-- Cardinal.ord_isLimit
theorem ord_isLimit {c} (co : ℵ₀ ≤ c) : (ord c).IsLimit 
Arguments: {c : Cardinal.{u_1}} (co : ℵ₀ ≤ c)
Goal: c.ord.IsLimit
-- Cardinal.noMaxOrder
theorem noMaxOrder {c} (h : ℵ₀ ≤ c) : NoMaxOrder c.ord.out.α 
Arguments: {c : Cardinal.{u_1}} (h : ℵ₀ ≤ c)
Goal: NoMaxOrder (Quotient.out c.ord).α
-- Cardinal.alephIdx.initialSeg_coe
@[simp]
theorem alephIdx.initialSeg_coe : (alephIdx.initialSeg : Cardinal → Ordinal) = alephIdx 
Goal: ⇑Cardinal.alephIdx.initialSeg = Cardinal.alephIdx
-- Cardinal.alephIdx_lt
@[simp]
theorem alephIdx_lt {a b} : alephIdx a < alephIdx b ↔ a < b 
Arguments: {a b : Cardinal.{u_1}}
Goal: a.alephIdx < b.alephIdx ↔ a < b
-- Cardinal.alephIdx_le
@[simp]
theorem alephIdx_le {a b} : alephIdx a ≤ alephIdx b ↔ a ≤ b 
Arguments: {a b : Cardinal.{u_1}}
Goal: a.alephIdx ≤ b.alephIdx ↔ a ≤ b