ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContDiffWithinAt.mul
theorem ContDiffWithinAt.mul {s : Set E} {f g : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x * g x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {s : Set E} {f g : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => f x * g x) s x
-- ContDiffAt.mul
nonrec theorem ContDiffAt.mul {f g : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) :
    ContDiffAt 𝕜 n (fun x => f x * g x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f g : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x)
Goal: ContDiffAt 𝕜 n (fun x => f x * g x) x
-- ContDiffOn.mul
theorem ContDiffOn.mul {f g : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
    ContDiffOn 𝕜 n (fun x => f x * g x) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f g : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s)
Goal: ContDiffOn 𝕜 n (fun x => f x * g x) s
-- ContDiff.mul
theorem ContDiff.mul {f g : E → 𝔸} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n fun x => f x * g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f g : E → 𝔸} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n fun x => f x * g x
-- contDiffWithinAt_prod'
theorem contDiffWithinAt_prod' {t : Finset ι} {f : ι → E → 𝔸'}
    (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x) : ContDiffWithinAt 𝕜 n (∏ i ∈ t, f i) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x)
Goal: ContDiffWithinAt 𝕜 n (∏ i ∈ t, f i) s x
-- contDiffWithinAt_prod
theorem contDiffWithinAt_prod {t : Finset ι} {f : ι → E → 𝔸'}
    (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x) :
    ContDiffWithinAt 𝕜 n (fun y => ∏ i ∈ t, f i y) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffWithinAt 𝕜 n (f i) s x)
Goal: ContDiffWithinAt 𝕜 n (fun y => ∏ i ∈ t, f i y) s x
-- contDiffAt_prod'
theorem contDiffAt_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x) :
    ContDiffAt 𝕜 n (∏ i ∈ t, f i) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x)
Goal: ContDiffAt 𝕜 n (∏ i ∈ t, f i) x
-- contDiffAt_prod
theorem contDiffAt_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x) :
    ContDiffAt 𝕜 n (fun y => ∏ i ∈ t, f i y) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffAt 𝕜 n (f i) x)
Goal: ContDiffAt 𝕜 n (fun y => ∏ i ∈ t, f i y) x
-- contDiffOn_prod'
theorem contDiffOn_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s) :
    ContDiffOn 𝕜 n (∏ i ∈ t, f i) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s)
Goal: ContDiffOn 𝕜 n (∏ i ∈ t, f i) s
-- contDiffOn_prod
theorem contDiffOn_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s) :
    ContDiffOn 𝕜 n (fun y => ∏ i ∈ t, f i y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiffOn 𝕜 n (f i) s)
Goal: ContDiffOn 𝕜 n (fun y => ∏ i ∈ t, f i y) s
-- contDiff_prod'
theorem contDiff_prod' {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i)) :
    ContDiff 𝕜 n (∏ i ∈ t, f i) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i))
Goal: ContDiff 𝕜 n (∏ i ∈ t, f i)
-- contDiff_prod
theorem contDiff_prod {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i)) :
    ContDiff 𝕜 n fun y => ∏ i ∈ t, f i y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {𝔸' : Type u_4} {ι : Type u_5} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {t : Finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, ContDiff 𝕜 n (f i))
Goal: ContDiff 𝕜 n fun y => ∏ i ∈ t, f i y
-- ContDiff.pow
theorem ContDiff.pow {f : E → 𝔸} (hf : ContDiff 𝕜 n f) : ∀ m : ℕ, ContDiff 𝕜 n fun x => f x ^ m
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : E → 𝔸} (hf : ContDiff 𝕜 n f) (m : ℕ)
Goal: ContDiff 𝕜 n fun x => f x ^ m
-- ContDiffWithinAt.pow
theorem ContDiffWithinAt.pow {f : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x) (m : ℕ) :
    ContDiffWithinAt 𝕜 n (fun y => f y ^ m) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : E → 𝔸} (hf : ContDiffWithinAt 𝕜 n f s x) (m : ℕ)
Goal: ContDiffWithinAt 𝕜 n (fun y => f y ^ m) s x
-- ContDiffAt.pow
nonrec theorem ContDiffAt.pow {f : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (m : ℕ) :
    ContDiffAt 𝕜 n (fun y => f y ^ m) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : E → 𝔸} (hf : ContDiffAt 𝕜 n f x) (m : ℕ)
Goal: ContDiffAt 𝕜 n (fun y => f y ^ m) x
-- ContDiffOn.pow
theorem ContDiffOn.pow {f : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (m : ℕ) :
    ContDiffOn 𝕜 n (fun y => f y ^ m) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {n : ℕ∞} {𝔸 : Type u_3} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {f : E → 𝔸} (hf : ContDiffOn 𝕜 n f s) (m : ℕ)
Goal: ContDiffOn 𝕜 n (fun y => f y ^ m) s
-- ContDiffWithinAt.div_const
theorem ContDiffWithinAt.div_const {f : E → 𝕜'} {n} (hf : ContDiffWithinAt 𝕜 n f s x) (c : 𝕜') :
    ContDiffWithinAt 𝕜 n (fun x => f x / c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {𝕜' : Type u_6} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n f s x) (c : 𝕜')
Goal: ContDiffWithinAt 𝕜 n (fun x => f x / c) s x
-- ContDiffAt.div_const
nonrec theorem ContDiffAt.div_const {f : E → 𝕜'} {n} (hf : ContDiffAt 𝕜 n f x) (c : 𝕜') :
    ContDiffAt 𝕜 n (fun x => f x / c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝕜' : Type u_6} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffAt 𝕜 n f x) (c : 𝕜')
Goal: ContDiffAt 𝕜 n (fun x => f x / c) x
-- ContDiffOn.div_const
theorem ContDiffOn.div_const {f : E → 𝕜'} {n} (hf : ContDiffOn 𝕜 n f s) (c : 𝕜') :
    ContDiffOn 𝕜 n (fun x => f x / c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝕜' : Type u_6} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (c : 𝕜')
Goal: ContDiffOn 𝕜 n (fun x => f x / c) s
-- ContDiff.div_const
theorem ContDiff.div_const {f : E → 𝕜'} {n} (hf : ContDiff 𝕜 n f) (c : 𝕜') :
    ContDiff 𝕜 n fun x => f x / c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_6} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiff 𝕜 n f) (c : 𝕜')
Goal: ContDiff 𝕜 n fun x => f x / c
-- contDiff_smul
theorem contDiff_smul : ContDiff 𝕜 n fun p : 𝕜 × F => p.1 • p.2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞}
Goal: ContDiff 𝕜 n fun p => p.1 • p.2
-- ContDiffWithinAt.smul
theorem ContDiffWithinAt.smul {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) : ContDiffWithinAt 𝕜 n (fun x => f x • g x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x)
Goal: ContDiffWithinAt 𝕜 n (fun x => f x • g x) s x
-- ContDiffAt.smul
theorem ContDiffAt.smul {f : E → 𝕜} {g : E → F} (hf : ContDiffAt 𝕜 n f x)
    (hg : ContDiffAt 𝕜 n g x) : ContDiffAt 𝕜 n (fun x => f x • g x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {n : ℕ∞} {f : E → 𝕜} {g : E → F} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x)
Goal: ContDiffAt 𝕜 n (fun x => f x • g x) x
-- ContDiff.smul
theorem ContDiff.smul {f : E → 𝕜} {g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n fun x => f x • g x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {f : E → 𝕜} {g : E → F} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n fun x => f x • g x
-- ContDiffOn.smul
theorem ContDiffOn.smul {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffOn 𝕜 n f s)
    (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x => f x • g x) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {s : Set E} {f : E → 𝕜} {g : E → F} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s)
Goal: ContDiffOn 𝕜 n (fun x => f x • g x) s
-- contDiff_const_smul
theorem contDiff_const_smul (c : R) : ContDiff 𝕜 n fun p : F => c • p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R)
Goal: ContDiff 𝕜 n fun p => c • p
-- ContDiffWithinAt.const_smul
theorem ContDiffWithinAt.const_smul {s : Set E} {f : E → F} {x : E} (c : R)
    (hf : ContDiffWithinAt 𝕜 n f s x) : ContDiffWithinAt 𝕜 n (fun y => c • f y) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {s : Set E} {f : E → F} {x : E} (c : R) (hf : ContDiffWithinAt 𝕜 n f s x)
Goal: ContDiffWithinAt 𝕜 n (fun y => c • f y) s x
-- ContDiffAt.const_smul
theorem ContDiffAt.const_smul {f : E → F} {x : E} (c : R) (hf : ContDiffAt 𝕜 n f x) :
    ContDiffAt 𝕜 n (fun y => c • f y) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {f : E → F} {x : E} (c : R) (hf : ContDiffAt 𝕜 n f x)
Goal: ContDiffAt 𝕜 n (fun y => c • f y) x
-- ContDiff.const_smul
theorem ContDiff.const_smul {f : E → F} (c : R) (hf : ContDiff 𝕜 n f) :
    ContDiff 𝕜 n fun y => c • f y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {f : E → F} (c : R) (hf : ContDiff 𝕜 n f)
Goal: ContDiff 𝕜 n fun y => c • f y
-- ContDiffOn.const_smul
theorem ContDiffOn.const_smul {s : Set E} {f : E → F} (c : R) (hf : ContDiffOn 𝕜 n f s) :
    ContDiffOn 𝕜 n (fun y => c • f y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {s : Set E} {f : E → F} (c : R) (hf : ContDiffOn 𝕜 n f s)
Goal: ContDiffOn 𝕜 n (fun y => c • f y) s
-- iteratedFDerivWithin_const_smul_apply
theorem iteratedFDerivWithin_const_smul_apply (hf : ContDiffOn 𝕜 i f s) (hu : UniqueDiffOn 𝕜 s)
    (hx : x ∈ s) : iteratedFDerivWithin 𝕜 i (a • f) s x = a • iteratedFDerivWithin 𝕜 i f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {i : ℕ} {a : R} (hf : ContDiffOn 𝕜 (↑i) f s) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
Goal: iteratedFDerivWithin 𝕜 i (a • f) s x = a • iteratedFDerivWithin 𝕜 i f s x
-- iteratedFDeriv_const_smul_apply
theorem iteratedFDeriv_const_smul_apply {x : E} (hf : ContDiff 𝕜 i f) :
    iteratedFDeriv 𝕜 i (a • f) x = a • iteratedFDeriv 𝕜 i f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {i : ℕ} {a : R} {x : E} (hf : ContDiff 𝕜 (↑i) f)
Goal: iteratedFDeriv 𝕜 i (a • f) x = a • iteratedFDeriv 𝕜 i f x
-- iteratedFDeriv_const_smul_apply'
theorem iteratedFDeriv_const_smul_apply' {x : E} (hf : ContDiff 𝕜 i f) :
    iteratedFDeriv 𝕜 i (fun x ↦ a • f x) x = a • iteratedFDeriv 𝕜 i f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {i : ℕ} {a : R} {x : E} (hf : ContDiff 𝕜 (↑i) f)
Goal: iteratedFDeriv 𝕜 i (fun x => a • f x) x = a • iteratedFDeriv 𝕜 i f x
-- ContDiffWithinAt.prod_map'
theorem ContDiffWithinAt.prod_map' {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {p : E × E'}
    (hf : ContDiffWithinAt 𝕜 n f s p.1) (hg : ContDiffWithinAt 𝕜 n g t p.2) :
    ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {p : E × E'} (hf : ContDiffWithinAt 𝕜 n f s p.1) (hg : ContDiffWithinAt 𝕜 n g t p.2)
Goal: ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) p
-- ContDiffWithinAt.prod_map
theorem ContDiffWithinAt.prod_map {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {x : E}
    {y : E'} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g t y) :
    ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) (x, y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g t y)
Goal: ContDiffWithinAt 𝕜 n (Prod.map f g) (s ×ˢ t) (x, y)
-- ContDiffOn.prod_map
theorem ContDiffOn.prod_map {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type*}
    [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'}
    (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g t) : ContDiffOn 𝕜 n (Prod.map f g) (s ×ˢ t) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_6} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {s : Set E} {t : Set E'} {f : E → F} {g : E' → F'} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g t)
Goal: ContDiffOn 𝕜 n (Prod.map f g) (s ×ˢ t)
-- ContDiffAt.prod_map
theorem ContDiffAt.prod_map {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffAt 𝕜 n f x)
    (hg : ContDiffAt 𝕜 n g y) : ContDiffAt 𝕜 n (Prod.map f g) (x, y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g y)
Goal: ContDiffAt 𝕜 n (Prod.map f g) (x, y)
-- ContDiffAt.prod_map'
theorem ContDiffAt.prod_map' {f : E → F} {g : E' → F'} {p : E × E'} (hf : ContDiffAt 𝕜 n f p.1)
    (hg : ContDiffAt 𝕜 n g p.2) : ContDiffAt 𝕜 n (Prod.map f g) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {f : E → F} {g : E' → F'} {p : E × E'} (hf : ContDiffAt 𝕜 n f p.1) (hg : ContDiffAt 𝕜 n g p.2)
Goal: ContDiffAt 𝕜 n (Prod.map f g) p
-- ContDiff.prod_map
theorem ContDiff.prod_map {f : E → F} {g : E' → F'} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
    ContDiff 𝕜 n (Prod.map f g) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F' : Type u_4} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {f : E → F} {g : E' → F'} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n (Prod.map f g)
-- contDiff_prod_mk_left
theorem contDiff_prod_mk_left (f₀ : F) : ContDiff 𝕜 n fun e : E => (e, f₀) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (f₀ : F)
Goal: ContDiff 𝕜 n fun e => (e, f₀)
-- contDiff_prod_mk_right
theorem contDiff_prod_mk_right (e₀ : E) : ContDiff 𝕜 n fun f : F => (e₀, f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ∞} (e₀ : E)
Goal: ContDiff 𝕜 n fun f => (e₀, f)
-- contDiffAt_ring_inverse
theorem contDiffAt_ring_inverse [CompleteSpace R] (x : Rˣ) :
    ContDiffAt 𝕜 n Ring.inverse (x : R) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {n : ℕ∞} {R : Type u_3} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R] (x : Rˣ)
Goal: ContDiffAt 𝕜 n Ring.inverse ↑x
-- contDiffAt_inv
theorem contDiffAt_inv {x : 𝕜'} (hx : x ≠ 0) {n} : ContDiffAt 𝕜 n Inv.inv x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {x : 𝕜'} (hx : x ≠ 0) {n : ℕ∞}
Goal: ContDiffAt 𝕜 n Inv.inv x
-- contDiffOn_inv
theorem contDiffOn_inv {n} : ContDiffOn 𝕜 n (Inv.inv : 𝕜' → 𝕜') {0}ᶜ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {n : ℕ∞}
Goal: ContDiffOn 𝕜 n Inv.inv {0}ᶜ
-- ContDiffWithinAt.inv
theorem ContDiffWithinAt.inv {f : E → 𝕜'} {n} (hf : ContDiffWithinAt 𝕜 n f s x) (hx : f x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => (f x)⁻¹) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n f s x) (hx : f x ≠ 0)
Goal: ContDiffWithinAt 𝕜 n (fun x => (f x)⁻¹) s x
-- ContDiffOn.inv
theorem ContDiffOn.inv {f : E → 𝕜'} {n} (hf : ContDiffOn 𝕜 n f s) (h : ∀ x ∈ s, f x ≠ 0) :
    ContDiffOn 𝕜 n (fun x => (f x)⁻¹) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (h : ∀ x ∈ s, f x ≠ 0)
Goal: ContDiffOn 𝕜 n (fun x => (f x)⁻¹) s
-- ContDiffAt.inv
nonrec theorem ContDiffAt.inv {f : E → 𝕜'} {n} (hf : ContDiffAt 𝕜 n f x) (hx : f x ≠ 0) :
    ContDiffAt 𝕜 n (fun x => (f x)⁻¹) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiffAt 𝕜 n f x) (hx : f x ≠ 0)
Goal: ContDiffAt 𝕜 n (fun x => (f x)⁻¹) x
-- ContDiff.inv
theorem ContDiff.inv {f : E → 𝕜'} {n} (hf : ContDiff 𝕜 n f) (h : ∀ x, f x ≠ 0) :
    ContDiff 𝕜 n fun x => (f x)⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_4} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [CompleteSpace 𝕜'] {f : E → 𝕜'} {n : ℕ∞} (hf : ContDiff 𝕜 n f) (h : ∀ (x : E), f x ≠ 0)
Goal: ContDiff 𝕜 n fun x => (f x)⁻¹
-- ContDiffWithinAt.div
theorem ContDiffWithinAt.div [CompleteSpace 𝕜] {f g : E → 𝕜} {n} (hf : ContDiffWithinAt 𝕜 n f s x)
    (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0) :
    ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E} [CompleteSpace 𝕜] {f g : E → 𝕜} {n : ℕ∞} (hf : ContDiffWithinAt 𝕜 n f s x) (hg : ContDiffWithinAt 𝕜 n g s x) (hx : g x ≠ 0)
Goal: ContDiffWithinAt 𝕜 n (fun x => f x / g x) s x
-- ContDiffOn.div
theorem ContDiffOn.div [CompleteSpace 𝕜] {f g : E → 𝕜} {n} (hf : ContDiffOn 𝕜 n f s)
    (hg : ContDiffOn 𝕜 n g s) (h₀ : ∀ x ∈ s, g x ≠ 0) : ContDiffOn 𝕜 n (f / g) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} [CompleteSpace 𝕜] {f g : E → 𝕜} {n : ℕ∞} (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: ContDiffOn 𝕜 n (f / g) s
-- ContDiffAt.div
nonrec theorem ContDiffAt.div [CompleteSpace 𝕜] {f g : E → 𝕜} {n} (hf : ContDiffAt 𝕜 n f x)
    (hg : ContDiffAt 𝕜 n g x) (hx : g x ≠ 0) : ContDiffAt 𝕜 n (fun x => f x / g x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} [CompleteSpace 𝕜] {f g : E → 𝕜} {n : ℕ∞} (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) (hx : g x ≠ 0)
Goal: ContDiffAt 𝕜 n (fun x => f x / g x) x