ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasDerivAt.const_smul
nonrec theorem HasDerivAt.const_smul (c : R) (hf : HasDerivAt f f' x) :
    HasDerivAt (fun y => c • f y) (c • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R) (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun y => c • f y) (c • f') x
-- derivWithin_const_smul
theorem derivWithin_const_smul (hxs : UniqueDiffWithinAt 𝕜 s x) (c : R)
    (hf : DifferentiableWithinAt 𝕜 f s x) :
    derivWithin (fun y => c • f y) s x = c • derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (hxs : UniqueDiffWithinAt 𝕜 s x) (c : R) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: derivWithin (fun y => c • f y) s x = c • derivWithin f s x
-- deriv_const_smul
theorem deriv_const_smul (c : R) (hf : DifferentiableAt 𝕜 f x) :
    deriv (fun y => c • f y) x = c • deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R) (hf : DifferentiableAt 𝕜 f x)
Goal: deriv (fun y => c • f y) x = c • deriv f x
-- deriv_const_smul'
lemma deriv_const_smul' {f : 𝕜 → F} {x : 𝕜} {R : Type*} [Field R] [Module R F] [SMulCommClass 𝕜 R F]
    [ContinuousConstSMul R F] (c : R) :
    deriv (fun y ↦ c • f y) x = c • deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {R : Type u_3} [Field R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R)
Goal: deriv (fun y => c • f y) x = c • deriv f x
-- HasDerivWithinAt.mul
theorem HasDerivWithinAt.mul (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x) :
    HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸} {c' d' : 𝔸} (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x)
Goal: HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x
-- HasDerivAt.mul
theorem HasDerivAt.mul (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) :
    HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸} {c' d' : 𝔸} (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x)
Goal: HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x
-- HasStrictDerivAt.mul
theorem HasStrictDerivAt.mul (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
    HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸} {c' d' : 𝔸} (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x)
Goal: HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x
-- derivWithin_mul
theorem derivWithin_mul (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (hd : DifferentiableWithinAt 𝕜 d s x) :
    derivWithin (fun y => c y * d y) s x = derivWithin c s x * d x + c x * derivWithin d s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x)
Goal: derivWithin (fun y => c y * d y) s x = derivWithin c s x * d x + c x * derivWithin d s x
-- deriv_mul
@[simp]
theorem deriv_mul (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) :
    deriv (fun y => c y * d y) x = deriv c x * d x + c x * deriv d x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c d : 𝕜 → 𝔸} (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x)
Goal: deriv (fun y => c y * d y) x = deriv c x * d x + c x * deriv d x
-- HasDerivWithinAt.mul_const
theorem HasDerivWithinAt.mul_const (hc : HasDerivWithinAt c c' s x) (d : 𝔸) :
    HasDerivWithinAt (fun y => c y * d) (c' * d) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} {c' : 𝔸} (hc : HasDerivWithinAt c c' s x) (d : 𝔸)
Goal: HasDerivWithinAt (fun y => c y * d) (c' * d) s x
-- HasDerivAt.mul_const
theorem HasDerivAt.mul_const (hc : HasDerivAt c c' x) (d : 𝔸) :
    HasDerivAt (fun y => c y * d) (c' * d) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} {c' : 𝔸} (hc : HasDerivAt c c' x) (d : 𝔸)
Goal: HasDerivAt (fun y => c y * d) (c' * d) x
-- hasDerivAt_mul_const
theorem hasDerivAt_mul_const (c : 𝕜) : HasDerivAt (fun x => x * c) c x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} (c : 𝕜)
Goal: HasDerivAt (fun x => x * c) c x
-- HasStrictDerivAt.mul_const
theorem HasStrictDerivAt.mul_const (hc : HasStrictDerivAt c c' x) (d : 𝔸) :
    HasStrictDerivAt (fun y => c y * d) (c' * d) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} {c' : 𝔸} (hc : HasStrictDerivAt c c' x) (d : 𝔸)
Goal: HasStrictDerivAt (fun y => c y * d) (c' * d) x
-- derivWithin_mul_const
theorem derivWithin_mul_const (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (d : 𝔸) : derivWithin (fun y => c y * d) s x = derivWithin c s x * d 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝔸)
Goal: derivWithin (fun y => c y * d) s x = derivWithin c s x * d
-- deriv_mul_const
theorem deriv_mul_const (hc : DifferentiableAt 𝕜 c x) (d : 𝔸) :
    deriv (fun y => c y * d) x = deriv c x * d 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {c : 𝕜 → 𝔸} (hc : DifferentiableAt 𝕜 c x) (d : 𝔸)
Goal: deriv (fun y => c y * d) x = deriv c x * d
-- deriv_mul_const_field
theorem deriv_mul_const_field (v : 𝕜') : deriv (fun y => u y * v) x = deriv u x * v 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {u : 𝕜 → 𝕜'} (v : 𝕜')
Goal: deriv (fun y => u y * v) x = deriv u x * v
-- deriv_mul_const_field'
@[simp]
theorem deriv_mul_const_field' (v : 𝕜') : (deriv fun x => u x * v) = fun x => deriv u x * v 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {u : 𝕜 → 𝕜'} (v : 𝕜')
Goal: (deriv fun x => u x * v) = fun x => deriv u x * v
-- HasDerivWithinAt.const_mul
theorem HasDerivWithinAt.const_mul (c : 𝔸) (hd : HasDerivWithinAt d d' s x) :
    HasDerivWithinAt (fun y => c * d y) (c * d') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {d : 𝕜 → 𝔸} {d' : 𝔸} (c : 𝔸) (hd : HasDerivWithinAt d d' s x)
Goal: HasDerivWithinAt (fun y => c * d y) (c * d') s x
-- HasDerivAt.const_mul
theorem HasDerivAt.const_mul (c : 𝔸) (hd : HasDerivAt d d' x) :
    HasDerivAt (fun y => c * d y) (c * d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {d : 𝕜 → 𝔸} {d' : 𝔸} (c : 𝔸) (hd : HasDerivAt d d' x)
Goal: HasDerivAt (fun y => c * d y) (c * d') x
-- HasStrictDerivAt.const_mul
theorem HasStrictDerivAt.const_mul (c : 𝔸) (hd : HasStrictDerivAt d d' x) :
    HasStrictDerivAt (fun y => c * d y) (c * d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {d : 𝕜 → 𝔸} {d' : 𝔸} (c : 𝔸) (hd : HasStrictDerivAt d d' x)
Goal: HasStrictDerivAt (fun y => c * d y) (c * d') x
-- derivWithin_const_mul
theorem derivWithin_const_mul (hxs : UniqueDiffWithinAt 𝕜 s x) (c : 𝔸)
    (hd : DifferentiableWithinAt 𝕜 d s x) :
    derivWithin (fun y => c * d y) s x = c * derivWithin d s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {d : 𝕜 → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : 𝔸) (hd : DifferentiableWithinAt 𝕜 d s x)
Goal: derivWithin (fun y => c * d y) s x = c * derivWithin d s x
-- deriv_const_mul
theorem deriv_const_mul (c : 𝔸) (hd : DifferentiableAt 𝕜 d x) :
    deriv (fun y => c * d y) x = c * deriv d x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {d : 𝕜 → 𝔸} (c : 𝔸) (hd : DifferentiableAt 𝕜 d x)
Goal: deriv (fun y => c * d y) x = c * deriv d x
-- deriv_const_mul_field
theorem deriv_const_mul_field (u : 𝕜') : deriv (fun y => u * v y) x = u * deriv v x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {v : 𝕜 → 𝕜'} (u : 𝕜')
Goal: deriv (fun y => u * v y) x = u * deriv v x
-- deriv_const_mul_field'
@[simp]
theorem deriv_const_mul_field' (u : 𝕜') : (deriv fun x => u * v x) = fun x => u * deriv v x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {v : 𝕜 → 𝕜'} (u : 𝕜')
Goal: (deriv fun x => u * v x) = fun x => u * deriv v x
-- HasDerivAt.finset_prod
theorem HasDerivAt.finset_prod (hf : ∀ i ∈ u, HasDerivAt (f i) (f' i) x) :
    HasDerivAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_2} [DecidableEq ι] {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} {f' : ι → 𝔸'} (hf : ∀ i ∈ u, HasDerivAt (f i) (f' i) x)
Goal: HasDerivAt (fun x => ∏ i ∈ u, f i x) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x
-- HasDerivWithinAt.finset_prod
theorem HasDerivWithinAt.finset_prod (hf : ∀ i ∈ u, HasDerivWithinAt (f i) (f' i) s x) :
    HasDerivWithinAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_2} [DecidableEq ι] {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} {f' : ι → 𝔸'} (hf : ∀ i ∈ u, HasDerivWithinAt (f i) (f' i) s x)
Goal: HasDerivWithinAt (fun x => ∏ i ∈ u, f i x) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) s x
-- HasStrictDerivAt.finset_prod
theorem HasStrictDerivAt.finset_prod (hf : ∀ i ∈ u, HasStrictDerivAt (f i) (f' i) x) :
    HasStrictDerivAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_2} [DecidableEq ι] {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} {f' : ι → 𝔸'} (hf : ∀ i ∈ u, HasStrictDerivAt (f i) (f' i) x)
Goal: HasStrictDerivAt (fun x => ∏ i ∈ u, f i x) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x
-- deriv_finset_prod
theorem deriv_finset_prod (hf : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x) :
    deriv (∏ i ∈ u, f i ·) x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • deriv (f i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_2} [DecidableEq ι] {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hf : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x)
Goal: deriv (fun x => ∏ i ∈ u, f i x) x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • deriv (f i) x
-- derivWithin_finset_prod
theorem derivWithin_finset_prod (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) :
    derivWithin (∏ i ∈ u, f i ·) s x =
      ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_2} [DecidableEq ι] {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x)
Goal: derivWithin (fun x => ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x
-- DifferentiableAt.finset_prod
theorem DifferentiableAt.finset_prod (hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x) :
    DifferentiableAt 𝕜 (∏ i ∈ u, f i ·) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {ι : Type u_2} {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x)
Goal: DifferentiableAt 𝕜 (fun x => ∏ i ∈ u, f i x) x
-- DifferentiableWithinAt.finset_prod
theorem DifferentiableWithinAt.finset_prod (hd : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) :
    DifferentiableWithinAt 𝕜 (∏ i ∈ u, f i ·) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_2} {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hd : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x)
Goal: DifferentiableWithinAt 𝕜 (fun x => ∏ i ∈ u, f i x) s x
-- DifferentiableOn.finset_prod
theorem DifferentiableOn.finset_prod (hd : ∀ i ∈ u, DifferentiableOn 𝕜 (f i) s) :
    DifferentiableOn 𝕜 (∏ i ∈ u, f i ·) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} {ι : Type u_2} {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hd : ∀ i ∈ u, DifferentiableOn 𝕜 (f i) s)
Goal: DifferentiableOn 𝕜 (fun x => ∏ i ∈ u, f i x) s
-- Differentiable.finset_prod
theorem Differentiable.finset_prod (hd : ∀ i ∈ u, Differentiable 𝕜 (f i)) :
    Differentiable 𝕜 (∏ i ∈ u, f i ·) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {ι : Type u_2} {𝔸' : Type u_3} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'} (hd : ∀ i ∈ u, Differentiable 𝕜 (f i))
Goal: Differentiable 𝕜 fun x => ∏ i ∈ u, f i x
-- HasDerivAt.div_const
theorem HasDerivAt.div_const (hc : HasDerivAt c c' x) (d : 𝕜') :
    HasDerivAt (fun x => c x / d) (c' / d) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivAt c c' x) (d : 𝕜')
Goal: HasDerivAt (fun x => c x / d) (c' / d) x
-- HasDerivWithinAt.div_const
theorem HasDerivWithinAt.div_const (hc : HasDerivWithinAt c c' s x) (d : 𝕜') :
    HasDerivWithinAt (fun x => c x / d) (c' / d) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivWithinAt c c' s x) (d : 𝕜')
Goal: HasDerivWithinAt (fun x => c x / d) (c' / d) s x
-- HasStrictDerivAt.div_const
theorem HasStrictDerivAt.div_const (hc : HasStrictDerivAt c c' x) (d : 𝕜') :
    HasStrictDerivAt (fun x => c x / d) (c' / d) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasStrictDerivAt c c' x) (d : 𝕜')
Goal: HasStrictDerivAt (fun x => c x / d) (c' / d) x
-- DifferentiableWithinAt.div_const
theorem DifferentiableWithinAt.div_const (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝕜') :
    DifferentiableWithinAt 𝕜 (fun x => c x / d) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝕜')
Goal: DifferentiableWithinAt 𝕜 (fun x => c x / d) s x
-- DifferentiableAt.div_const
@[simp]
theorem DifferentiableAt.div_const (hc : DifferentiableAt 𝕜 c x) (d : 𝕜') :
    DifferentiableAt 𝕜 (fun x => c x / d) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (d : 𝕜')
Goal: DifferentiableAt 𝕜 (fun x => c x / d) x
-- DifferentiableOn.div_const
theorem DifferentiableOn.div_const (hc : DifferentiableOn 𝕜 c s) (d : 𝕜') :
    DifferentiableOn 𝕜 (fun x => c x / d) s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (hc : DifferentiableOn 𝕜 c s) (d : 𝕜')
Goal: DifferentiableOn 𝕜 (fun x => c x / d) s
-- Differentiable.div_const
@[simp]
theorem Differentiable.div_const (hc : Differentiable 𝕜 c) (d : 𝕜') :
    Differentiable 𝕜 fun x => c x / d 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (hc : Differentiable 𝕜 c) (d : 𝕜')
Goal: Differentiable 𝕜 fun x => c x / d
-- derivWithin_div_const
theorem derivWithin_div_const (hc : DifferentiableWithinAt 𝕜 c s x)
    (d : 𝕜') (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun x => c x / d) s x = derivWithin c s x / d 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝕜') (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => c x / d) s x = derivWithin c s x / d
-- deriv_div_const
@[simp]
theorem deriv_div_const (d : 𝕜') : deriv (fun x => c x / d) x = deriv c x / d 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {c : 𝕜 → 𝕜'} (d : 𝕜')
Goal: deriv (fun x => c x / d) x = deriv c x / d
-- HasStrictDerivAt.clm_comp
theorem HasStrictDerivAt.clm_comp (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
    HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {d : 𝕜 → E →L[𝕜] F} {d' : E →L[𝕜] F} (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x)
Goal: HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x
-- HasDerivWithinAt.clm_comp
theorem HasDerivWithinAt.clm_comp (hc : HasDerivWithinAt c c' s x)
    (hd : HasDerivWithinAt d d' s x) :
    HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : 𝕜} {s : Set 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {d : 𝕜 → E →L[𝕜] F} {d' : E →L[𝕜] F} (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x)
Goal: HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x
-- HasDerivAt.clm_comp
theorem HasDerivAt.clm_comp (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) :
    HasDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {d : 𝕜 → E →L[𝕜] F} {d' : E →L[𝕜] F} (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x)
Goal: HasDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x
-- derivWithin_clm_comp
theorem derivWithin_clm_comp (hc : DifferentiableWithinAt 𝕜 c s x)
    (hd : DifferentiableWithinAt 𝕜 d s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun y => (c y).comp (d y)) s x =
      (derivWithin c s x).comp (d x) + (c x).comp (derivWithin d s x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : 𝕜} {s : Set 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {d : 𝕜 → E →L[𝕜] F} (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun y => (c y).comp (d y)) s x = (derivWithin c s x).comp (d x) + (c x).comp (derivWithin d s x)
-- deriv_clm_comp
theorem deriv_clm_comp (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) :
    deriv (fun y => (c y).comp (d y)) x = (deriv c x).comp (d x) + (c x).comp (deriv d x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {d : 𝕜 → E →L[𝕜] F} (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x)
Goal: deriv (fun y => (c y).comp (d y)) x = (deriv c x).comp (d x) + (c x).comp (deriv d x)
-- HasStrictDerivAt.clm_apply
theorem HasStrictDerivAt.clm_apply (hc : HasStrictDerivAt c c' x) (hu : HasStrictDerivAt u u' x) :
    HasStrictDerivAt (fun y => (c y) (u y)) (c' (u x) + c x u') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {u : 𝕜 → F} {u' : F} (hc : HasStrictDerivAt c c' x) (hu : HasStrictDerivAt u u' x)
Goal: HasStrictDerivAt (fun y => (c y) (u y)) (c' (u x) + (c x) u') x
-- HasDerivWithinAt.clm_apply
theorem HasDerivWithinAt.clm_apply (hc : HasDerivWithinAt c c' s x)
    (hu : HasDerivWithinAt u u' s x) :
    HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + c x u') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {u : 𝕜 → F} {u' : F} (hc : HasDerivWithinAt c c' s x) (hu : HasDerivWithinAt u u' s x)
Goal: HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + (c x) u') s x
-- HasDerivAt.clm_apply
theorem HasDerivAt.clm_apply (hc : HasDerivAt c c' x) (hu : HasDerivAt u u' x) :
    HasDerivAt (fun y => (c y) (u y)) (c' (u x) + c x u') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G} {u : 𝕜 → F} {u' : F} (hc : HasDerivAt c c' x) (hu : HasDerivAt u u' x)
Goal: HasDerivAt (fun y => (c y) (u y)) (c' (u x) + (c x) u') x
-- derivWithin_clm_apply
theorem derivWithin_clm_apply (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (hu : DifferentiableWithinAt 𝕜 u s x) :
    derivWithin (fun y => (c y) (u y)) s x = derivWithin c s x (u x) + c x (derivWithin u s x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {u : 𝕜 → F} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hu : DifferentiableWithinAt 𝕜 u s x)
Goal: derivWithin (fun y => (c y) (u y)) s x = (derivWithin c s x) (u x) + (c x) (derivWithin u s x)
-- deriv_clm_apply
theorem deriv_clm_apply (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x) :
    deriv (fun y => (c y) (u y)) x = deriv c x (u x) + c x (deriv u x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {G : Type u_2} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {u : 𝕜 → F} (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x)
Goal: deriv (fun y => (c y) (u y)) x = (deriv c x) (u x) + (c x) (deriv u x)