ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- iteratedDerivWithin_add
theorem iteratedDerivWithin_add (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
    iteratedDerivWithin n (f + g) s x =
      iteratedDerivWithin n f s x + iteratedDerivWithin n g s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f g : 𝕜 → F} (hf : ContDiffOn 𝕜 (↑n) f s) (hg : ContDiffOn 𝕜 (↑n) g s)
Goal: iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x
-- iteratedDerivWithin_congr
theorem iteratedDerivWithin_congr (hfg : Set.EqOn f g s) :
    Set.EqOn (iteratedDerivWithin n f s) (iteratedDerivWithin n g s) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {s : Set 𝕜} (h : UniqueDiffOn 𝕜 s) {f g : 𝕜 → F} (hfg : Set.EqOn f g s)
Goal: Set.EqOn (iteratedDerivWithin n f s) (iteratedDerivWithin n g s) s
-- iteratedDerivWithin_const_add
theorem iteratedDerivWithin_const_add (hn : 0 < n) (c : F) :
    iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f : 𝕜 → F} (hn : 0 < n) (c : F)
Goal: iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x
-- iteratedDerivWithin_const_neg
theorem iteratedDerivWithin_const_neg (hn : 0 < n) (c : F) :
    iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f : 𝕜 → F} (hn : 0 < n) (c : F)
Goal: iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x
-- iteratedDerivWithin_const_smul
theorem iteratedDerivWithin_const_smul (c : R) (hf : ContDiffOn 𝕜 n f s) :
    iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {R : Type u_3} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f : 𝕜 → F} (c : R) (hf : ContDiffOn 𝕜 (↑n) f s)
Goal: iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x
-- iteratedDerivWithin_const_mul
theorem iteratedDerivWithin_const_mul (c : 𝕜) {f : 𝕜 → 𝕜} (hf : ContDiffOn 𝕜 n f s) :
    iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) (c : 𝕜) {f : 𝕜 → 𝕜} (hf : ContDiffOn 𝕜 (↑n) f s)
Goal: iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x
-- iteratedDerivWithin_neg
theorem iteratedDerivWithin_neg :
    iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) (f : 𝕜 → F)
Goal: iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x
-- iteratedDerivWithin_neg'
theorem iteratedDerivWithin_neg' :
    iteratedDerivWithin n (fun z => -f z) s x = -iteratedDerivWithin n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) (f : 𝕜 → F)
Goal: iteratedDerivWithin n (fun z => -f z) s x = -iteratedDerivWithin n f s x
-- iteratedDerivWithin_sub
theorem iteratedDerivWithin_sub (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
    iteratedDerivWithin n (f - g) s x =
      iteratedDerivWithin n f s x - iteratedDerivWithin n g s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f g : 𝕜 → F} (hf : ContDiffOn 𝕜 (↑n) f s) (hg : ContDiffOn 𝕜 (↑n) g s)
Goal: iteratedDerivWithin n (f - g) s x = iteratedDerivWithin n f s x - iteratedDerivWithin n g s x
-- iteratedDeriv_const_smul
theorem iteratedDeriv_const_smul {n : ℕ} {f : 𝕜 → F} (h : ContDiff 𝕜 n f) (c : 𝕜) :
    iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n • iteratedDeriv n f (c * x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : ℕ} {f : 𝕜 → F} (h : ContDiff 𝕜 (↑n) f) (c : 𝕜)
Goal: (iteratedDeriv n fun x => f (c * x)) = fun x => c ^ n • iteratedDeriv n f (c * x)
-- iteratedDeriv_const_mul
theorem iteratedDeriv_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 n f) (c : 𝕜) :
    iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 (↑n) f) (c : 𝕜)
Goal: (iteratedDeriv n fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x)
-- iteratedDeriv_neg
lemma iteratedDeriv_neg (n : ℕ) (f : 𝕜 → F) (a : 𝕜) :
    iteratedDeriv n (fun x ↦ -(f x)) a = -(iteratedDeriv n f a) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : ℕ) (f : 𝕜 → F) (a : 𝕜)
Goal: iteratedDeriv n (fun x => -f x) a = -iteratedDeriv n f a
-- iteratedDeriv_comp_neg
lemma iteratedDeriv_comp_neg (n : ℕ) (f : 𝕜 → F) (a : 𝕜) :
    iteratedDeriv n (fun x ↦ f (-x)) a = (-1 : 𝕜) ^ n • iteratedDeriv n f (-a) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : ℕ) (f : 𝕜 → F) (a : 𝕜)
Goal: iteratedDeriv n (fun x => f (-x)) a = (-1) ^ n • iteratedDeriv n f (-a)
-- logDeriv_apply
theorem logDeriv_apply (f : 𝕜 → 𝕜') (x : 𝕜) : logDeriv f x = deriv f x / f x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] (f : 𝕜 → 𝕜') (x : 𝕜)
Goal: logDeriv f x = deriv f x / f x
-- logDeriv_eq_zero_of_not_differentiableAt
lemma logDeriv_eq_zero_of_not_differentiableAt (f : 𝕜 → 𝕜') (x : 𝕜) (h : ¬DifferentiableAt 𝕜 f x) :
    logDeriv f x = 0 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] (f : 𝕜 → 𝕜') (x : 𝕜) (h : ¬DifferentiableAt 𝕜 f x)
Goal: logDeriv f x = 0
-- logDeriv_id
@[simp]
theorem logDeriv_id (x : 𝕜) : logDeriv id x = 1 / x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: logDeriv id x = 1 / x
-- logDeriv_id'
@[simp] theorem logDeriv_id' (x : 𝕜) : logDeriv (·) x = 1 / x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: logDeriv (fun x => x) x = 1 / x
-- logDeriv_const
@[simp]
theorem logDeriv_const (a : 𝕜') : logDeriv (fun _ : 𝕜 ↦ a) = 0 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] (a : 𝕜')
Goal: (logDeriv fun x => a) = 0
-- logDeriv_mul
theorem logDeriv_mul {f g : 𝕜 → 𝕜'} (x : 𝕜) (hf : f x ≠ 0) (hg : g x ≠ 0)
    (hdf : DifferentiableAt 𝕜 f x) (hdg : DifferentiableAt 𝕜 g x) :
      logDeriv (fun z => f z * g z) x = logDeriv f x + logDeriv g x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f g : 𝕜 → 𝕜'} (x : 𝕜) (hf : f x ≠ 0) (hg : g x ≠ 0) (hdf : DifferentiableAt 𝕜 f x) (hdg : DifferentiableAt 𝕜 g x)
Goal: logDeriv (fun z => f z * g z) x = logDeriv f x + logDeriv g x
-- logDeriv_mul_const
theorem logDeriv_mul_const {f : 𝕜 → 𝕜'} (x : 𝕜) (a : 𝕜') (ha : a ≠ 0):
    logDeriv (fun z => f z * a) x = logDeriv f x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : 𝕜 → 𝕜'} (x : 𝕜) (a : 𝕜') (ha : a ≠ 0)
Goal: logDeriv (fun z => f z * a) x = logDeriv f x
-- logDeriv_const_mul
theorem logDeriv_const_mul {f : 𝕜 → 𝕜'} (x : 𝕜) (a : 𝕜') (ha : a ≠ 0):
    logDeriv (fun z => a * f z) x = logDeriv f x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : 𝕜 → 𝕜'} (x : 𝕜) (a : 𝕜') (ha : a ≠ 0)
Goal: logDeriv (fun z => a * f z) x = logDeriv f x
-- logDeriv_prod
theorem logDeriv_prod {ι : Type*} (s : Finset ι) (f : ι → 𝕜 → 𝕜') (x : 𝕜) (hf : ∀ i ∈ s, f i x ≠ 0)
    (hd : ∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) :
    logDeriv (∏ i ∈ s, f i ·) x = ∑ i ∈ s, logDeriv (f i) x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {ι : Type u_3} (s : Finset ι) (f : ι → 𝕜 → 𝕜') (x : 𝕜) (hf : ∀ i ∈ s, f i x ≠ 0) (hd : ∀ i ∈ s, DifferentiableAt 𝕜 (f i) x)
Goal: logDeriv (fun x => ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x
-- logDeriv_fun_zpow
lemma logDeriv_fun_zpow {f : 𝕜 → 𝕜'} {x : 𝕜} (hdf : DifferentiableAt 𝕜 f x) (n : ℤ) :
    logDeriv (f · ^ n) x = n * logDeriv f x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : 𝕜 → 𝕜'} {x : 𝕜} (hdf : DifferentiableAt 𝕜 f x) (n : ℤ)
Goal: logDeriv (fun x => f x ^ n) x = ↑n * logDeriv f x
-- logDeriv_fun_pow
lemma logDeriv_fun_pow {f : 𝕜 → 𝕜'} {x : 𝕜} (hdf : DifferentiableAt 𝕜 f x) (n : ℕ) :
    logDeriv (f · ^ n) x = n * logDeriv f x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : 𝕜 → 𝕜'} {x : 𝕜} (hdf : DifferentiableAt 𝕜 f x) (n : ℕ)
Goal: logDeriv (fun x => f x ^ n) x = ↑n * logDeriv f x
-- logDeriv_zpow
@[simp]
lemma logDeriv_zpow (x : 𝕜) (n : ℤ) : logDeriv (· ^ n) x = n / x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜) (n : ℤ)
Goal: logDeriv (fun x => x ^ n) x = ↑n / x
-- logDeriv_pow
@[simp]
lemma logDeriv_pow (x : 𝕜) (n : ℕ) : logDeriv (· ^ n) x = n / x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜) (n : ℕ)
Goal: logDeriv (fun x => x ^ n) x = ↑n / x
-- logDeriv_inv
@[simp] lemma logDeriv_inv (x : 𝕜) : logDeriv (·⁻¹) x = -1 / x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: logDeriv (fun x => x⁻¹) x = -1 / x
-- logDeriv_comp
theorem logDeriv_comp {f : 𝕜' → 𝕜'} {g : 𝕜 → 𝕜'} {x : 𝕜} (hf : DifferentiableAt 𝕜' f (g x))
    (hg : DifferentiableAt 𝕜 g x) : logDeriv (f ∘ g) x = logDeriv f (g x) * deriv g x 
Arguments: {𝕜 : Type u_1} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {f : 𝕜' → 𝕜'} {g : 𝕜 → 𝕜'} {x : 𝕜} (hf : DifferentiableAt 𝕜' f (g x)) (hg : DifferentiableAt 𝕜 g x)
Goal: logDeriv (f ∘ g) x = logDeriv f (g x) * deriv g x
-- hofer
theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
    {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
    ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' 
Arguments: {X : Type u_1} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε) {ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ (y : X), 0 ≤ ϕ y)
Goal: ∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'
-- HasStrictFDerivAt.prod
protected theorem HasStrictFDerivAt.prod (hf₁ : HasStrictFDerivAt f₁ f₁' x)
    (hf₂ : HasStrictFDerivAt f₂ f₂' x) :
    HasStrictFDerivAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {f₁' : E →L[𝕜] F} {x : E} {f₂ : E → G} {f₂' : E →L[𝕜] G} (hf₁ : HasStrictFDerivAt f₁ f₁' x) (hf₂ : HasStrictFDerivAt f₂ f₂' x)
Goal: HasStrictFDerivAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x
-- HasFDerivAtFilter.prod
theorem HasFDerivAtFilter.prod (hf₁ : HasFDerivAtFilter f₁ f₁' x L)
    (hf₂ : HasFDerivAtFilter f₂ f₂' x L) :
    HasFDerivAtFilter (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {f₁' : E →L[𝕜] F} {x : E} {L : Filter E} {f₂ : E → G} {f₂' : E →L[𝕜] G} (hf₁ : HasFDerivAtFilter f₁ f₁' x L) (hf₂ : HasFDerivAtFilter f₂ f₂' x L)
Goal: HasFDerivAtFilter (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x L
-- HasFDerivWithinAt.prod
@[fun_prop]
nonrec theorem HasFDerivWithinAt.prod (hf₁ : HasFDerivWithinAt f₁ f₁' s x)
    (hf₂ : HasFDerivWithinAt f₂ f₂' s x) :
    HasFDerivWithinAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {f₁' : E →L[𝕜] F} {x : E} {s : Set E} {f₂ : E → G} {f₂' : E →L[𝕜] G} (hf₁ : HasFDerivWithinAt f₁ f₁' s x) (hf₂ : HasFDerivWithinAt f₂ f₂' s x)
Goal: HasFDerivWithinAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') s x
-- HasFDerivAt.prod
@[fun_prop]
nonrec theorem HasFDerivAt.prod (hf₁ : HasFDerivAt f₁ f₁' x) (hf₂ : HasFDerivAt f₂ f₂' x) :
    HasFDerivAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {f₁' : E →L[𝕜] F} {x : E} {f₂ : E → G} {f₂' : E →L[𝕜] G} (hf₁ : HasFDerivAt f₁ f₁' x) (hf₂ : HasFDerivAt f₂ f₂' x)
Goal: HasFDerivAt (fun x => (f₁ x, f₂ x)) (f₁'.prod f₂') x
-- hasFDerivAt_prod_mk_left
@[fun_prop]
theorem hasFDerivAt_prod_mk_left (e₀ : E) (f₀ : F) :
    HasFDerivAt (fun e : E => (e, f₀)) (inl 𝕜 E F) e₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e₀ : E) (f₀ : F)
Goal: HasFDerivAt (fun e => (e, f₀)) (ContinuousLinearMap.inl 𝕜 E F) e₀
-- hasFDerivAt_prod_mk_right
@[fun_prop]
theorem hasFDerivAt_prod_mk_right (e₀ : E) (f₀ : F) :
    HasFDerivAt (fun f : F => (e₀, f)) (inr 𝕜 E F) f₀ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e₀ : E) (f₀ : F)
Goal: HasFDerivAt (fun f => (e₀, f)) (ContinuousLinearMap.inr 𝕜 E F) f₀
-- DifferentiableWithinAt.prod
@[fun_prop]
theorem DifferentiableWithinAt.prod (hf₁ : DifferentiableWithinAt 𝕜 f₁ s x)
    (hf₂ : DifferentiableWithinAt 𝕜 f₂ s x) :
    DifferentiableWithinAt 𝕜 (fun x : E => (f₁ x, f₂ x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {x : E} {s : Set E} {f₂ : E → G} (hf₁ : DifferentiableWithinAt 𝕜 f₁ s x) (hf₂ : DifferentiableWithinAt 𝕜 f₂ s x)
Goal: DifferentiableWithinAt 𝕜 (fun x => (f₁ x, f₂ x)) s x
-- DifferentiableAt.prod
@[simp, fun_prop]
theorem DifferentiableAt.prod (hf₁ : DifferentiableAt 𝕜 f₁ x) (hf₂ : DifferentiableAt 𝕜 f₂ x) :
    DifferentiableAt 𝕜 (fun x : E => (f₁ x, f₂ x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {x : E} {f₂ : E → G} (hf₁ : DifferentiableAt 𝕜 f₁ x) (hf₂ : DifferentiableAt 𝕜 f₂ x)
Goal: DifferentiableAt 𝕜 (fun x => (f₁ x, f₂ x)) x
-- DifferentiableOn.prod
@[fun_prop]
theorem DifferentiableOn.prod (hf₁ : DifferentiableOn 𝕜 f₁ s) (hf₂ : DifferentiableOn 𝕜 f₂ s) :
    DifferentiableOn 𝕜 (fun x : E => (f₁ x, f₂ x)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {s : Set E} {f₂ : E → G} (hf₁ : DifferentiableOn 𝕜 f₁ s) (hf₂ : DifferentiableOn 𝕜 f₂ s)
Goal: DifferentiableOn 𝕜 (fun x => (f₁ x, f₂ x)) s
-- Differentiable.prod
@[simp, fun_prop]
theorem Differentiable.prod (hf₁ : Differentiable 𝕜 f₁) (hf₂ : Differentiable 𝕜 f₂) :
    Differentiable 𝕜 fun x : E => (f₁ x, f₂ x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {f₂ : E → G} (hf₁ : Differentiable 𝕜 f₁) (hf₂ : Differentiable 𝕜 f₂)
Goal: Differentiable 𝕜 fun x => (f₁ x, f₂ x)
-- DifferentiableAt.fderiv_prod
theorem DifferentiableAt.fderiv_prod (hf₁ : DifferentiableAt 𝕜 f₁ x)
    (hf₂ : DifferentiableAt 𝕜 f₂ x) :
    fderiv 𝕜 (fun x : E => (f₁ x, f₂ x)) x = (fderiv 𝕜 f₁ x).prod (fderiv 𝕜 f₂ x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {x : E} {f₂ : E → G} (hf₁ : DifferentiableAt 𝕜 f₁ x) (hf₂ : DifferentiableAt 𝕜 f₂ x)
Goal: fderiv 𝕜 (fun x => (f₁ x, f₂ x)) x = (fderiv 𝕜 f₁ x).prod (fderiv 𝕜 f₂ x)
-- DifferentiableWithinAt.fderivWithin_prod
theorem DifferentiableWithinAt.fderivWithin_prod (hf₁ : DifferentiableWithinAt 𝕜 f₁ s x)
    (hf₂ : DifferentiableWithinAt 𝕜 f₂ s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x : E => (f₁ x, f₂ x)) s x =
      (fderivWithin 𝕜 f₁ s x).prod (fderivWithin 𝕜 f₂ s x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₁ : E → F} {x : E} {s : Set E} {f₂ : E → G} (hf₁ : DifferentiableWithinAt 𝕜 f₁ s x) (hf₂ : DifferentiableWithinAt 𝕜 f₂ s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => (f₁ x, f₂ x)) s x = (fderivWithin 𝕜 f₁ s x).prod (fderivWithin 𝕜 f₂ s x)
-- hasStrictFDerivAt_fst
@[fun_prop]
theorem hasStrictFDerivAt_fst : HasStrictFDerivAt (@Prod.fst E F) (fst 𝕜 E F) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F}
Goal: HasStrictFDerivAt Prod.fst (ContinuousLinearMap.fst 𝕜 E F) p
-- HasStrictFDerivAt.fst
@[fun_prop]
protected theorem HasStrictFDerivAt.fst (h : HasStrictFDerivAt f₂ f₂' x) :
    HasStrictFDerivAt (fun x => (f₂ x).1) ((fst 𝕜 F G).comp f₂') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasStrictFDerivAt f₂ f₂' x)
Goal: HasStrictFDerivAt (fun x => (f₂ x).1) ((ContinuousLinearMap.fst 𝕜 F G).comp f₂') x
-- hasFDerivAtFilter_fst
theorem hasFDerivAtFilter_fst {L : Filter (E × F)} :
    HasFDerivAtFilter (@Prod.fst E F) (fst 𝕜 E F) p L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F} {L : Filter (E × F)}
Goal: HasFDerivAtFilter Prod.fst (ContinuousLinearMap.fst 𝕜 E F) p L
-- HasFDerivAtFilter.fst
protected theorem HasFDerivAtFilter.fst (h : HasFDerivAtFilter f₂ f₂' x L) :
    HasFDerivAtFilter (fun x => (f₂ x).1) ((fst 𝕜 F G).comp f₂') x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {L : Filter E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivAtFilter f₂ f₂' x L)
Goal: HasFDerivAtFilter (fun x => (f₂ x).1) ((ContinuousLinearMap.fst 𝕜 F G).comp f₂') x L
-- hasFDerivAt_fst
@[fun_prop]
theorem hasFDerivAt_fst : HasFDerivAt (@Prod.fst E F) (fst 𝕜 E F) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F}
Goal: HasFDerivAt Prod.fst (ContinuousLinearMap.fst 𝕜 E F) p
-- HasFDerivAt.fst
@[fun_prop]
protected nonrec theorem HasFDerivAt.fst (h : HasFDerivAt f₂ f₂' x) :
    HasFDerivAt (fun x => (f₂ x).1) ((fst 𝕜 F G).comp f₂') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivAt f₂ f₂' x)
Goal: HasFDerivAt (fun x => (f₂ x).1) ((ContinuousLinearMap.fst 𝕜 F G).comp f₂') x
-- hasFDerivWithinAt_fst
@[fun_prop]
theorem hasFDerivWithinAt_fst {s : Set (E × F)} :
    HasFDerivWithinAt (@Prod.fst E F) (fst 𝕜 E F) s p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F} {s : Set (E × F)}
Goal: HasFDerivWithinAt Prod.fst (ContinuousLinearMap.fst 𝕜 E F) s p
-- HasFDerivWithinAt.fst
@[fun_prop]
protected nonrec theorem HasFDerivWithinAt.fst (h : HasFDerivWithinAt f₂ f₂' s x) :
    HasFDerivWithinAt (fun x => (f₂ x).1) ((fst 𝕜 F G).comp f₂') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivWithinAt f₂ f₂' s x)
Goal: HasFDerivWithinAt (fun x => (f₂ x).1) ((ContinuousLinearMap.fst 𝕜 F G).comp f₂') s x
-- differentiableAt_fst
@[fun_prop]
theorem differentiableAt_fst : DifferentiableAt 𝕜 Prod.fst p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F}
Goal: DifferentiableAt 𝕜 Prod.fst p
-- DifferentiableAt.fst
@[simp, fun_prop]
protected theorem DifferentiableAt.fst (h : DifferentiableAt 𝕜 f₂ x) :
    DifferentiableAt 𝕜 (fun x => (f₂ x).1) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} (h : DifferentiableAt 𝕜 f₂ x)
Goal: DifferentiableAt 𝕜 (fun x => (f₂ x).1) x
-- differentiable_fst
@[fun_prop]
theorem differentiable_fst : Differentiable 𝕜 (Prod.fst : E × F → E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: Differentiable 𝕜 Prod.fst
-- Differentiable.fst
@[simp, fun_prop]
protected theorem Differentiable.fst (h : Differentiable 𝕜 f₂) :
    Differentiable 𝕜 fun x => (f₂ x).1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : E → F × G} (h : Differentiable 𝕜 f₂)
Goal: Differentiable 𝕜 fun x => (f₂ x).1
-- differentiableWithinAt_fst
@[fun_prop]
theorem differentiableWithinAt_fst {s : Set (E × F)} : DifferentiableWithinAt 𝕜 Prod.fst s p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F} {s : Set (E × F)}
Goal: DifferentiableWithinAt 𝕜 Prod.fst s p