ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasFDerivWithinAt.mul
@[fun_prop]
theorem HasFDerivWithinAt.mul (hc : HasFDerivWithinAt c c' s x) (hd : HasFDerivWithinAt d d' s x) :
    HasFDerivWithinAt (fun y => c y * d y) (c x • d' + d x • c') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c d : E → 𝔸'} {c' d' : E →L[𝕜] 𝔸'} (hc : HasFDerivWithinAt c c' s x) (hd : HasFDerivWithinAt d d' s x)
Goal: HasFDerivWithinAt (fun y => c y * d y) (c x • d' + d x • c') s x
-- HasFDerivAt.mul'
@[fun_prop]
theorem HasFDerivAt.mul' (ha : HasFDerivAt a a' x) (hb : HasFDerivAt b b' x) :
    HasFDerivAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} {a' b' : E →L[𝕜] 𝔸} (ha : HasFDerivAt a a' x) (hb : HasFDerivAt b b' x)
Goal: HasFDerivAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) x
-- HasFDerivAt.mul
@[fun_prop]
theorem HasFDerivAt.mul (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :
    HasFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c d : E → 𝔸'} {c' d' : E →L[𝕜] 𝔸'} (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x)
Goal: HasFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x
-- DifferentiableWithinAt.mul
@[fun_prop]
theorem DifferentiableWithinAt.mul (ha : DifferentiableWithinAt 𝕜 a s x)
    (hb : DifferentiableWithinAt 𝕜 b s x) : DifferentiableWithinAt 𝕜 (fun y => a y * b y) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (ha : DifferentiableWithinAt 𝕜 a s x) (hb : DifferentiableWithinAt 𝕜 b s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => a y * b y) s x
-- DifferentiableAt.mul
@[simp, fun_prop]
theorem DifferentiableAt.mul (ha : DifferentiableAt 𝕜 a x) (hb : DifferentiableAt 𝕜 b x) :
    DifferentiableAt 𝕜 (fun y => a y * b y) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (hb : DifferentiableAt 𝕜 b x)
Goal: DifferentiableAt 𝕜 (fun y => a y * b y) x
-- DifferentiableOn.mul
@[fun_prop]
theorem DifferentiableOn.mul (ha : DifferentiableOn 𝕜 a s) (hb : DifferentiableOn 𝕜 b s) :
    DifferentiableOn 𝕜 (fun y => a y * b y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (ha : DifferentiableOn 𝕜 a s) (hb : DifferentiableOn 𝕜 b s)
Goal: DifferentiableOn 𝕜 (fun y => a y * b y) s
-- Differentiable.mul
@[simp, fun_prop]
theorem Differentiable.mul (ha : Differentiable 𝕜 a) (hb : Differentiable 𝕜 b) :
    Differentiable 𝕜 fun y => a y * b y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (ha : Differentiable 𝕜 a) (hb : Differentiable 𝕜 b)
Goal: Differentiable 𝕜 fun y => a y * b y
-- DifferentiableWithinAt.pow
@[fun_prop]
theorem DifferentiableWithinAt.pow (ha : DifferentiableWithinAt 𝕜 a s x) :
    ∀ n : ℕ, DifferentiableWithinAt 𝕜 (fun x => a x ^ n) s x
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableWithinAt 𝕜 a s x) (n : ℕ)
Goal: DifferentiableWithinAt 𝕜 (fun x => a x ^ n) s x
-- DifferentiableAt.pow
@[simp, fun_prop]
theorem DifferentiableAt.pow (ha : DifferentiableAt 𝕜 a x) (n : ℕ) :
    DifferentiableAt 𝕜 (fun x => a x ^ n) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (n : ℕ)
Goal: DifferentiableAt 𝕜 (fun x => a x ^ n) x
-- DifferentiableOn.pow
@[fun_prop]
theorem DifferentiableOn.pow (ha : DifferentiableOn 𝕜 a s) (n : ℕ) :
    DifferentiableOn 𝕜 (fun x => a x ^ n) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableOn 𝕜 a s) (n : ℕ)
Goal: DifferentiableOn 𝕜 (fun x => a x ^ n) s
-- Differentiable.pow
@[simp, fun_prop]
theorem Differentiable.pow (ha : Differentiable 𝕜 a) (n : ℕ) : Differentiable 𝕜 fun x => a x ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : Differentiable 𝕜 a) (n : ℕ)
Goal: Differentiable 𝕜 fun x => a x ^ n
-- fderivWithin_mul'
theorem fderivWithin_mul' (hxs : UniqueDiffWithinAt 𝕜 s x) (ha : DifferentiableWithinAt 𝕜 a s x)
    (hb : DifferentiableWithinAt 𝕜 b s x) :
    fderivWithin 𝕜 (fun y => a y * b y) s x =
      a x • fderivWithin 𝕜 b s x + (fderivWithin 𝕜 a s x).smulRight (b x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (ha : DifferentiableWithinAt 𝕜 a s x) (hb : DifferentiableWithinAt 𝕜 b s x)
Goal: fderivWithin 𝕜 (fun y => a y * b y) s x = a x • fderivWithin 𝕜 b s x + (fderivWithin 𝕜 a s x).smulRight (b x)
-- fderivWithin_mul
theorem fderivWithin_mul (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (hd : DifferentiableWithinAt 𝕜 d s x) :
    fderivWithin 𝕜 (fun y => c y * d y) s x =
      c x • fderivWithin 𝕜 d s x + d x • fderivWithin 𝕜 c s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c d : E → 𝔸'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hd : DifferentiableWithinAt 𝕜 d s x)
Goal: fderivWithin 𝕜 (fun y => c y * d y) s x = c x • fderivWithin 𝕜 d s x + d x • fderivWithin 𝕜 c s x
-- fderiv_mul'
theorem fderiv_mul' (ha : DifferentiableAt 𝕜 a x) (hb : DifferentiableAt 𝕜 b x) :
    fderiv 𝕜 (fun y => a y * b y) x = a x • fderiv 𝕜 b x + (fderiv 𝕜 a x).smulRight (b x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (hb : DifferentiableAt 𝕜 b x)
Goal: fderiv 𝕜 (fun y => a y * b y) x = a x • fderiv 𝕜 b x + (fderiv 𝕜 a x).smulRight (b x)
-- fderiv_mul
theorem fderiv_mul (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x) :
    fderiv 𝕜 (fun y => c y * d y) x = c x • fderiv 𝕜 d x + d x • fderiv 𝕜 c x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c d : E → 𝔸'} (hc : DifferentiableAt 𝕜 c x) (hd : DifferentiableAt 𝕜 d x)
Goal: fderiv 𝕜 (fun y => c y * d y) x = c x • fderiv 𝕜 d x + d x • fderiv 𝕜 c x
-- HasStrictFDerivAt.mul_const'
@[fun_prop]
theorem HasStrictFDerivAt.mul_const' (ha : HasStrictFDerivAt a a' x) (b : 𝔸) :
    HasStrictFDerivAt (fun y => a y * b) (a'.smulRight b) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasStrictFDerivAt a a' x) (b : 𝔸)
Goal: HasStrictFDerivAt (fun y => a y * b) (a'.smulRight b) x
-- HasStrictFDerivAt.mul_const
@[fun_prop]
theorem HasStrictFDerivAt.mul_const (hc : HasStrictFDerivAt c c' x) (d : 𝔸') :
    HasStrictFDerivAt (fun y => c y * d) (d • c') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c : E → 𝔸'} {c' : E →L[𝕜] 𝔸'} (hc : HasStrictFDerivAt c c' x) (d : 𝔸')
Goal: HasStrictFDerivAt (fun y => c y * d) (d • c') x
-- HasFDerivWithinAt.mul_const'
@[fun_prop]
theorem HasFDerivWithinAt.mul_const' (ha : HasFDerivWithinAt a a' s x) (b : 𝔸) :
    HasFDerivWithinAt (fun y => a y * b) (a'.smulRight b) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasFDerivWithinAt a a' s x) (b : 𝔸)
Goal: HasFDerivWithinAt (fun y => a y * b) (a'.smulRight b) s x
-- HasFDerivWithinAt.mul_const
@[fun_prop]
theorem HasFDerivWithinAt.mul_const (hc : HasFDerivWithinAt c c' s x) (d : 𝔸') :
    HasFDerivWithinAt (fun y => c y * d) (d • c') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c : E → 𝔸'} {c' : E →L[𝕜] 𝔸'} (hc : HasFDerivWithinAt c c' s x) (d : 𝔸')
Goal: HasFDerivWithinAt (fun y => c y * d) (d • c') s x
-- HasFDerivAt.mul_const'
@[fun_prop]
theorem HasFDerivAt.mul_const' (ha : HasFDerivAt a a' x) (b : 𝔸) :
    HasFDerivAt (fun y => a y * b) (a'.smulRight b) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasFDerivAt a a' x) (b : 𝔸)
Goal: HasFDerivAt (fun y => a y * b) (a'.smulRight b) x
-- HasFDerivAt.mul_const
@[fun_prop]
theorem HasFDerivAt.mul_const (hc : HasFDerivAt c c' x) (d : 𝔸') :
    HasFDerivAt (fun y => c y * d) (d • c') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c : E → 𝔸'} {c' : E →L[𝕜] 𝔸'} (hc : HasFDerivAt c c' x) (d : 𝔸')
Goal: HasFDerivAt (fun y => c y * d) (d • c') x
-- DifferentiableWithinAt.mul_const
@[fun_prop]
theorem DifferentiableWithinAt.mul_const (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸) :
    DifferentiableWithinAt 𝕜 (fun y => a y * b) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸)
Goal: DifferentiableWithinAt 𝕜 (fun y => a y * b) s x
-- DifferentiableAt.mul_const
@[fun_prop]
theorem DifferentiableAt.mul_const (ha : DifferentiableAt 𝕜 a x) (b : 𝔸) :
    DifferentiableAt 𝕜 (fun y => a y * b) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (b : 𝔸)
Goal: DifferentiableAt 𝕜 (fun y => a y * b) x
-- DifferentiableOn.mul_const
@[fun_prop]
theorem DifferentiableOn.mul_const (ha : DifferentiableOn 𝕜 a s) (b : 𝔸) :
    DifferentiableOn 𝕜 (fun y => a y * b) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableOn 𝕜 a s) (b : 𝔸)
Goal: DifferentiableOn 𝕜 (fun y => a y * b) s
-- Differentiable.mul_const
@[fun_prop]
theorem Differentiable.mul_const (ha : Differentiable 𝕜 a) (b : 𝔸) :
    Differentiable 𝕜 fun y => a y * b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : Differentiable 𝕜 a) (b : 𝔸)
Goal: Differentiable 𝕜 fun y => a y * b
-- fderivWithin_mul_const'
theorem fderivWithin_mul_const' (hxs : UniqueDiffWithinAt 𝕜 s x)
    (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸) :
    fderivWithin 𝕜 (fun y => a y * b) s x = (fderivWithin 𝕜 a s x).smulRight b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸)
Goal: fderivWithin 𝕜 (fun y => a y * b) s x = (fderivWithin 𝕜 a s x).smulRight b
-- fderivWithin_mul_const
theorem fderivWithin_mul_const (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝔸') :
    fderivWithin 𝕜 (fun y => c y * d) s x = d • fderivWithin 𝕜 c s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c : E → 𝔸'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝔸')
Goal: fderivWithin 𝕜 (fun y => c y * d) s x = d • fderivWithin 𝕜 c s x
-- fderiv_mul_const'
theorem fderiv_mul_const' (ha : DifferentiableAt 𝕜 a x) (b : 𝔸) :
    fderiv 𝕜 (fun y => a y * b) x = (fderiv 𝕜 a x).smulRight b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (b : 𝔸)
Goal: fderiv 𝕜 (fun y => a y * b) x = (fderiv 𝕜 a x).smulRight b
-- fderiv_mul_const
theorem fderiv_mul_const (hc : DifferentiableAt 𝕜 c x) (d : 𝔸') :
    fderiv 𝕜 (fun y => c y * d) x = d • fderiv 𝕜 c x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c : E → 𝔸'} (hc : DifferentiableAt 𝕜 c x) (d : 𝔸')
Goal: fderiv 𝕜 (fun y => c y * d) x = d • fderiv 𝕜 c x
-- HasStrictFDerivAt.const_mul
@[fun_prop]
theorem HasStrictFDerivAt.const_mul (ha : HasStrictFDerivAt a a' x) (b : 𝔸) :
    HasStrictFDerivAt (fun y => b * a y) (b • a') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasStrictFDerivAt a a' x) (b : 𝔸)
Goal: HasStrictFDerivAt (fun y => b * a y) (b • a') x
-- HasFDerivWithinAt.const_mul
@[fun_prop]
theorem HasFDerivWithinAt.const_mul (ha : HasFDerivWithinAt a a' s x) (b : 𝔸) :
    HasFDerivWithinAt (fun y => b * a y) (b • a') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasFDerivWithinAt a a' s x) (b : 𝔸)
Goal: HasFDerivWithinAt (fun y => b * a y) (b • a') s x
-- HasFDerivAt.const_mul
@[fun_prop]
theorem HasFDerivAt.const_mul (ha : HasFDerivAt a a' x) (b : 𝔸) :
    HasFDerivAt (fun y => b * a y) (b • a') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} {a' : E →L[𝕜] 𝔸} (ha : HasFDerivAt a a' x) (b : 𝔸)
Goal: HasFDerivAt (fun y => b * a y) (b • a') x
-- DifferentiableWithinAt.const_mul
@[fun_prop]
theorem DifferentiableWithinAt.const_mul (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸) :
    DifferentiableWithinAt 𝕜 (fun y => b * a y) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸)
Goal: DifferentiableWithinAt 𝕜 (fun y => b * a y) s x
-- DifferentiableAt.const_mul
@[fun_prop]
theorem DifferentiableAt.const_mul (ha : DifferentiableAt 𝕜 a x) (b : 𝔸) :
    DifferentiableAt 𝕜 (fun y => b * a y) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (b : 𝔸)
Goal: DifferentiableAt 𝕜 (fun y => b * a y) x
-- DifferentiableOn.const_mul
@[fun_prop]
theorem DifferentiableOn.const_mul (ha : DifferentiableOn 𝕜 a s) (b : 𝔸) :
    DifferentiableOn 𝕜 (fun y => b * a y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableOn 𝕜 a s) (b : 𝔸)
Goal: DifferentiableOn 𝕜 (fun y => b * a y) s
-- Differentiable.const_mul
@[fun_prop]
theorem Differentiable.const_mul (ha : Differentiable 𝕜 a) (b : 𝔸) :
    Differentiable 𝕜 fun y => b * a y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : Differentiable 𝕜 a) (b : 𝔸)
Goal: Differentiable 𝕜 fun y => b * a y
-- fderivWithin_const_mul
theorem fderivWithin_const_mul (hxs : UniqueDiffWithinAt 𝕜 s x)
    (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸) :
    fderivWithin 𝕜 (fun y => b * a y) s x = b • fderivWithin 𝕜 a s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (hxs : UniqueDiffWithinAt 𝕜 s x) (ha : DifferentiableWithinAt 𝕜 a s x) (b : 𝔸)
Goal: fderivWithin 𝕜 (fun y => b * a y) s x = b • fderivWithin 𝕜 a s x
-- fderiv_const_mul
theorem fderiv_const_mul (ha : DifferentiableAt 𝕜 a x) (b : 𝔸) :
    fderiv 𝕜 (fun y => b * a y) x = b • fderiv 𝕜 a x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a : E → 𝔸} (ha : DifferentiableAt 𝕜 a x) (b : 𝔸)
Goal: fderiv 𝕜 (fun y => b * a y) x = b • fderiv 𝕜 a x
-- hasStrictFDerivAt_list_prod'
@[fun_prop]
theorem hasStrictFDerivAt_list_prod' [Fintype ι] {l : List ι} {x : ι → 𝔸} :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.map x).prod)
      (∑ i : Fin l.length, ((l.take i).map x).prod •
        smulRight (proj (l.get i)) ((l.drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] [Fintype ι] {l : List ι} {x : ι → 𝔸}
Goal: HasStrictFDerivAt (fun x => (List.map x l).prod)
  (∑ i : Fin l.length,
    (List.map x (List.take (↑i) l)).prod •
      (ContinuousLinearMap.proj (l.get i)).smulRight (List.map x (List.drop (↑i).succ l)).prod)
  x
-- hasStrictFDerivAt_list_prod_finRange'
@[fun_prop]
theorem hasStrictFDerivAt_list_prod_finRange' {n : ℕ} {x : Fin n → 𝔸} :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ ((List.finRange n).map x).prod)
      (∑ i : Fin n, (((List.finRange n).take i).map x).prod •
        smulRight (proj i) (((List.finRange n).drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {n : ℕ} {x : Fin n → 𝔸}
Goal: HasStrictFDerivAt (fun x => (List.map x (List.finRange n)).prod)
  (∑ i : Fin n,
    (List.map x (List.take (↑i) (List.finRange n))).prod •
      (ContinuousLinearMap.proj i).smulRight (List.map x (List.drop (↑i).succ (List.finRange n))).prod)
  x
-- hasStrictFDerivAt_list_prod_attach'
@[fun_prop]
theorem hasStrictFDerivAt_list_prod_attach' [DecidableEq ι] {l : List ι} {x : {i // i ∈ l} → 𝔸} :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.attach.map x).prod)
      (∑ i : Fin l.length, ((l.attach.take i).map x).prod •
        smulRight (proj (l.attach.get (i.cast l.length_attach.symm)))
          ((l.attach.drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] [DecidableEq ι] {l : List ι} {x : { i // i ∈ l } → 𝔸}
Goal: HasStrictFDerivAt (fun x => (List.map x l.attach).prod)
  (∑ i : Fin l.length,
    (List.map x (List.take (↑i) l.attach)).prod •
      (ContinuousLinearMap.proj (l.attach.get (Fin.cast ⋯ i))).smulRight
        (List.map x (List.drop (↑i).succ l.attach)).prod)
  x
-- hasFDerivAt_list_prod'
@[fun_prop]
theorem hasFDerivAt_list_prod' [Fintype ι] {l : List ι} {x : ι → 𝔸'} :
    HasFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.map x).prod)
      (∑ i : Fin l.length, ((l.take i).map x).prod •
        smulRight (proj (l.get i)) ((l.drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] [Fintype ι] {l : List ι} {x : ι → 𝔸'}
Goal: HasFDerivAt (fun x => (List.map x l).prod)
  (∑ i : Fin l.length,
    (List.map x (List.take (↑i) l)).prod •
      (ContinuousLinearMap.proj (l.get i)).smulRight (List.map x (List.drop (↑i).succ l)).prod)
  x
-- hasFDerivAt_list_prod_finRange'
@[fun_prop]
theorem hasFDerivAt_list_prod_finRange' {n : ℕ} {x : Fin n → 𝔸} :
    HasFDerivAt (𝕜 := 𝕜) (fun x ↦ ((List.finRange n).map x).prod)
      (∑ i : Fin n, (((List.finRange n).take i).map x).prod •
        smulRight (proj i) (((List.finRange n).drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {n : ℕ} {x : Fin n → 𝔸}
Goal: HasFDerivAt (fun x => (List.map x (List.finRange n)).prod)
  (∑ i : Fin n,
    (List.map x (List.take (↑i) (List.finRange n))).prod •
      (ContinuousLinearMap.proj i).smulRight (List.map x (List.drop (↑i).succ (List.finRange n))).prod)
  x
-- hasFDerivAt_list_prod_attach'
@[fun_prop]
theorem hasFDerivAt_list_prod_attach' [DecidableEq ι] {l : List ι} {x : {i // i ∈ l} → 𝔸} :
    HasFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.attach.map x).prod)
      (∑ i : Fin l.length, ((l.attach.take i).map x).prod •
        smulRight (proj (l.attach.get (i.cast l.length_attach.symm)))
          ((l.attach.drop (.succ i)).map x).prod) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸 : Type u_7} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] [DecidableEq ι] {l : List ι} {x : { i // i ∈ l } → 𝔸}
Goal: HasFDerivAt (fun x => (List.map x l.attach).prod)
  (∑ i : Fin l.length,
    (List.map x (List.take (↑i) l.attach)).prod •
      (ContinuousLinearMap.proj (l.attach.get (Fin.cast ⋯ i))).smulRight
        (List.map x (List.drop (↑i).succ l.attach)).prod)
  x
-- hasStrictFDerivAt_list_prod
@[fun_prop]
theorem hasStrictFDerivAt_list_prod [DecidableEq ι] [Fintype ι] {l : List ι} {x : ι → 𝔸'} :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.map x).prod)
      (l.map fun i ↦ ((l.erase i).map x).prod • proj i).sum x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] [DecidableEq ι] [Fintype ι] {l : List ι} {x : ι → 𝔸'}
Goal: HasStrictFDerivAt (fun x => (List.map x l).prod)
  (List.map (fun i => (List.map x (l.erase i)).prod • ContinuousLinearMap.proj i) l).sum x
-- hasStrictFDerivAt_multiset_prod
@[fun_prop]
theorem hasStrictFDerivAt_multiset_prod [DecidableEq ι] [Fintype ι] {u : Multiset ι} {x : ι → 𝔸'} :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (u.map x).prod)
      (u.map (fun i ↦ ((u.erase i).map x).prod • proj i)).sum x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] [DecidableEq ι] [Fintype ι] {u : Multiset ι} {x : ι → 𝔸'}
Goal: HasStrictFDerivAt (fun x => (Multiset.map x u).prod)
  (Multiset.map (fun i => (Multiset.map x (u.erase i)).prod • ContinuousLinearMap.proj i) u).sum x
-- hasFDerivAt_multiset_prod
@[fun_prop]
theorem hasFDerivAt_multiset_prod [DecidableEq ι] [Fintype ι] {u : Multiset ι} {x : ι → 𝔸'} :
    HasFDerivAt (𝕜 := 𝕜) (fun x ↦ (u.map x).prod)
      (Multiset.sum (u.map (fun i ↦ ((u.erase i).map x).prod • proj i))) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} {𝔸' : Type u_8} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] [DecidableEq ι] [Fintype ι] {u : Multiset ι} {x : ι → 𝔸'}
Goal: HasFDerivAt (fun x => (Multiset.map x u).prod)
  (Multiset.map (fun i => (Multiset.map x (u.erase i)).prod • ContinuousLinearMap.proj i) u).sum x