ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.differentiableAt_sinh
theorem differentiableAt_sinh : DifferentiableAt ℝ sinh x 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.sinh x
-- Real.deriv_sinh
@[simp]
theorem deriv_sinh : deriv sinh = cosh 
Goal: deriv Real.sinh = Real.cosh
-- Real.hasStrictDerivAt_cosh
theorem hasStrictDerivAt_cosh (x : ℝ) : HasStrictDerivAt cosh (sinh x) x 
Arguments: (x : ℝ)
Goal: HasStrictDerivAt Real.cosh (Real.sinh x) x
-- Real.hasDerivAt_cosh
theorem hasDerivAt_cosh (x : ℝ) : HasDerivAt cosh (sinh x) x 
Arguments: (x : ℝ)
Goal: HasDerivAt Real.cosh (Real.sinh x) x
-- Real.contDiff_cosh
theorem contDiff_cosh {n} : ContDiff ℝ n cosh 
Arguments: {n : ℕ∞}
Goal: ContDiff ℝ n Real.cosh
-- Real.differentiable_cosh
theorem differentiable_cosh : Differentiable ℝ cosh 
Goal: Differentiable ℝ Real.cosh
-- Real.differentiableAt_cosh
theorem differentiableAt_cosh : DifferentiableAt ℝ cosh x 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.cosh x
-- Real.deriv_cosh
@[simp]
theorem deriv_cosh : deriv cosh = sinh 
Goal: deriv Real.cosh = Real.sinh
-- Real.sinh_strictMono
theorem sinh_strictMono : StrictMono sinh 
Goal: StrictMono Real.sinh
-- Real.sinh_injective
theorem sinh_injective : Function.Injective sinh 
Goal: Function.Injective Real.sinh
-- Real.sinh_inj
@[simp]
theorem sinh_inj : sinh x = sinh y ↔ x = y 
Arguments: {x y : ℝ}
Goal: Real.sinh x = Real.sinh y ↔ x = y
-- Real.sinh_le_sinh
@[simp]
theorem sinh_le_sinh : sinh x ≤ sinh y ↔ x ≤ y 
Arguments: {x y : ℝ}
Goal: Real.sinh x ≤ Real.sinh y ↔ x ≤ y
-- Real.sinh_lt_sinh
@[simp]
theorem sinh_lt_sinh : sinh x < sinh y ↔ x < y 
Arguments: {x y : ℝ}
Goal: Real.sinh x < Real.sinh y ↔ x < y
-- Real.sinh_eq_zero
@[simp] lemma sinh_eq_zero : sinh x = 0 ↔ x = 0 
Arguments: {x : ℝ}
Goal: Real.sinh x = 0 ↔ x = 0
-- Real.sinh_ne_zero
lemma sinh_ne_zero : sinh x ≠ 0 ↔ x ≠ 0 
Arguments: {x : ℝ}
Goal: Real.sinh x ≠ 0 ↔ x ≠ 0
-- Real.sinh_pos_iff
@[simp]
theorem sinh_pos_iff : 0 < sinh x ↔ 0 < x 
Arguments: {x : ℝ}
Goal: 0 < Real.sinh x ↔ 0 < x
-- Real.sinh_nonpos_iff
@[simp]
theorem sinh_nonpos_iff : sinh x ≤ 0 ↔ x ≤ 0 
Arguments: {x : ℝ}
Goal: Real.sinh x ≤ 0 ↔ x ≤ 0
-- Real.sinh_neg_iff
@[simp]
theorem sinh_neg_iff : sinh x < 0 ↔ x < 0 
Arguments: {x : ℝ}
Goal: Real.sinh x < 0 ↔ x < 0
-- Real.sinh_nonneg_iff
@[simp]
theorem sinh_nonneg_iff : 0 ≤ sinh x ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: 0 ≤ Real.sinh x ↔ 0 ≤ x
-- Real.abs_sinh
theorem abs_sinh (x : ℝ) : |sinh x| = sinh |x| 
Arguments: (x : ℝ)
Goal: |Real.sinh x| = Real.sinh |x|
-- Real.cosh_strictMonoOn
theorem cosh_strictMonoOn : StrictMonoOn cosh (Ici 0) 
Goal: StrictMonoOn Real.cosh (Set.Ici 0)
-- Real.cosh_le_cosh
@[simp]
theorem cosh_le_cosh : cosh x ≤ cosh y ↔ |x| ≤ |y| 
Arguments: {x y : ℝ}
Goal: Real.cosh x ≤ Real.cosh y ↔ |x| ≤ |y|
-- Real.cosh_lt_cosh
@[simp]
theorem cosh_lt_cosh : cosh x < cosh y ↔ |x| < |y| 
Arguments: {x y : ℝ}
Goal: Real.cosh x < Real.cosh y ↔ |x| < |y|
-- Real.one_le_cosh
@[simp]
theorem one_le_cosh (x : ℝ) : 1 ≤ cosh x 
Arguments: (x : ℝ)
Goal: 1 ≤ Real.cosh x
-- Real.one_lt_cosh
@[simp]
theorem one_lt_cosh : 1 < cosh x ↔ x ≠ 0 
Arguments: {x : ℝ}
Goal: 1 < Real.cosh x ↔ x ≠ 0
-- Real.sinh_sub_id_strictMono
theorem sinh_sub_id_strictMono : StrictMono fun x => sinh x - x 
Goal: StrictMono fun x => Real.sinh x - x
-- Real.self_le_sinh_iff
@[simp]
theorem self_le_sinh_iff : x ≤ sinh x ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: x ≤ Real.sinh x ↔ 0 ≤ x
-- Real.sinh_le_self_iff
@[simp]
theorem sinh_le_self_iff : sinh x ≤ x ↔ x ≤ 0 
Arguments: {x : ℝ}
Goal: Real.sinh x ≤ x ↔ x ≤ 0
-- Real.self_lt_sinh_iff
@[simp]
theorem self_lt_sinh_iff : x < sinh x ↔ 0 < x 
Arguments: {x : ℝ}
Goal: x < Real.sinh x ↔ 0 < x
-- Real.sinh_lt_self_iff
@[simp]
theorem sinh_lt_self_iff : sinh x < x ↔ x < 0 
Arguments: {x : ℝ}
Goal: Real.sinh x < x ↔ x < 0
-- HasStrictDerivAt.cos
theorem HasStrictDerivAt.cos (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x
-- HasDerivAt.cos
theorem HasDerivAt.cos (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') x
-- HasDerivWithinAt.cos
theorem HasDerivWithinAt.cos (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') s x
-- derivWithin_cos
theorem derivWithin_cos (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.cos (f x)) s x = -Real.sin (f x) * derivWithin f s x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.cos (f x)) s x = -Real.sin (f x) * derivWithin f s x
-- deriv_cos
@[simp]
theorem deriv_cos (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.cos (f x)) x = -Real.sin (f x) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.cos (f x)) x = -Real.sin (f x) * deriv f x
-- HasStrictDerivAt.sin
theorem HasStrictDerivAt.sin (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x
-- HasDerivAt.sin
theorem HasDerivAt.sin (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') x
-- HasDerivWithinAt.sin
theorem HasDerivWithinAt.sin (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) * f') s x
-- derivWithin_sin
theorem derivWithin_sin (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.sin (f x)) s x = Real.cos (f x) * derivWithin f s x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.sin (f x)) s x = Real.cos (f x) * derivWithin f s x
-- deriv_sin
@[simp]
theorem deriv_sin (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.sin (f x)) x = Real.cos (f x) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.sin (f x)) x = Real.cos (f x) * deriv f x
-- HasStrictDerivAt.cosh
theorem HasStrictDerivAt.cosh (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x
-- HasDerivAt.cosh
theorem HasDerivAt.cosh (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') x
-- HasDerivWithinAt.cosh
theorem HasDerivWithinAt.cosh (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.cosh (f x)) (Real.sinh (f x) * f') s x
-- derivWithin_cosh
theorem derivWithin_cosh (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.cosh (f x)) s x = Real.sinh (f x) * derivWithin f s x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.cosh (f x)) s x = Real.sinh (f x) * derivWithin f s x
-- deriv_cosh
@[simp]
theorem deriv_cosh (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.cosh (f x)) x = Real.sinh (f x) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.cosh (f x)) x = Real.sinh (f x) * deriv f x
-- HasStrictDerivAt.sinh
theorem HasStrictDerivAt.sinh (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x
-- HasDerivAt.sinh
theorem HasDerivAt.sinh (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') x
-- HasDerivWithinAt.sinh
theorem HasDerivWithinAt.sinh (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') s x 
Arguments: {f : ℝ → ℝ} {f' x : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => Real.sinh (f x)) (Real.cosh (f x) * f') s x
-- derivWithin_sinh
theorem derivWithin_sinh (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => Real.sinh (f x)) s x = Real.cosh (f x) * derivWithin f s x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.sinh (f x)) s x = Real.cosh (f x) * derivWithin f s x
-- deriv_sinh
@[simp]
theorem deriv_sinh (hc : DifferentiableAt ℝ f x) :
    deriv (fun x => Real.sinh (f x)) x = Real.cosh (f x) * deriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hc : DifferentiableAt ℝ f x)
Goal: deriv (fun x => Real.sinh (f x)) x = Real.cosh (f x) * deriv f x
-- HasStrictFDerivAt.cos
theorem HasStrictFDerivAt.cos (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') x
-- HasFDerivAt.cos
theorem HasFDerivAt.cos (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') x
-- HasFDerivWithinAt.cos
theorem HasFDerivWithinAt.cos (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) • f') s x
-- DifferentiableWithinAt.cos
theorem DifferentiableWithinAt.cos (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.cos (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x)
Goal: DifferentiableWithinAt ℝ (fun x => Real.cos (f x)) s x
-- DifferentiableAt.cos
@[simp]
theorem DifferentiableAt.cos (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.cos (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: DifferentiableAt ℝ (fun x => Real.cos (f x)) x
-- DifferentiableOn.cos
theorem DifferentiableOn.cos (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.cos (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s)
Goal: DifferentiableOn ℝ (fun x => Real.cos (f x)) s
-- Differentiable.cos
@[simp]
theorem Differentiable.cos (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.cos (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f)
Goal: Differentiable ℝ fun x => Real.cos (f x)
-- fderivWithin_cos
theorem fderivWithin_cos (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.cos (f x)) s x = -Real.sin (f x) • fderivWithin ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: fderivWithin ℝ (fun x => Real.cos (f x)) s x = -Real.sin (f x) • fderivWithin ℝ f s x
-- fderiv_cos
@[simp]
theorem fderiv_cos (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.cos (f x)) x = -Real.sin (f x) • fderiv ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: fderiv ℝ (fun x => Real.cos (f x)) x = -Real.sin (f x) • fderiv ℝ f x
-- ContDiff.cos
theorem ContDiff.cos {n} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.cos (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => Real.cos (f x)
-- ContDiffAt.cos
theorem ContDiffAt.cos {n} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.cos (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => Real.cos (f x)) x
-- ContDiffOn.cos
theorem ContDiffOn.cos {n} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.cos (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun x => Real.cos (f x)) s
-- ContDiffWithinAt.cos
theorem ContDiffWithinAt.cos {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.cos (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun x => Real.cos (f x)) s x
-- HasStrictFDerivAt.sin
theorem HasStrictFDerivAt.sin (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') x
-- HasFDerivAt.sin
theorem HasFDerivAt.sin (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') x
-- HasFDerivWithinAt.sin
theorem HasFDerivWithinAt.sin (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Real.sin (f x)) (Real.cos (f x) • f') s x
-- DifferentiableWithinAt.sin
theorem DifferentiableWithinAt.sin (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.sin (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x)
Goal: DifferentiableWithinAt ℝ (fun x => Real.sin (f x)) s x
-- DifferentiableAt.sin
@[simp]
theorem DifferentiableAt.sin (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.sin (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: DifferentiableAt ℝ (fun x => Real.sin (f x)) x
-- DifferentiableOn.sin
theorem DifferentiableOn.sin (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.sin (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s)
Goal: DifferentiableOn ℝ (fun x => Real.sin (f x)) s
-- Differentiable.sin
@[simp]
theorem Differentiable.sin (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.sin (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f)
Goal: Differentiable ℝ fun x => Real.sin (f x)
-- fderivWithin_sin
theorem fderivWithin_sin (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.sin (f x)) s x = Real.cos (f x) • fderivWithin ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: fderivWithin ℝ (fun x => Real.sin (f x)) s x = Real.cos (f x) • fderivWithin ℝ f s x
-- fderiv_sin
@[simp]
theorem fderiv_sin (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.sin (f x)) x = Real.cos (f x) • fderiv ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: fderiv ℝ (fun x => Real.sin (f x)) x = Real.cos (f x) • fderiv ℝ f x
-- ContDiff.sin
theorem ContDiff.sin {n} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.sin (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => Real.sin (f x)
-- ContDiffAt.sin
theorem ContDiffAt.sin {n} (hf : ContDiffAt ℝ n f x) : ContDiffAt ℝ n (fun x => Real.sin (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => Real.sin (f x)) x
-- ContDiffOn.sin
theorem ContDiffOn.sin {n} (hf : ContDiffOn ℝ n f s) : ContDiffOn ℝ n (fun x => Real.sin (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun x => Real.sin (f x)) s
-- ContDiffWithinAt.sin
theorem ContDiffWithinAt.sin {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.sin (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun x => Real.sin (f x)) s x
-- HasStrictFDerivAt.cosh
theorem HasStrictFDerivAt.cosh (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') x
-- HasFDerivAt.cosh
theorem HasFDerivAt.cosh (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') x
-- HasFDerivWithinAt.cosh
theorem HasFDerivWithinAt.cosh (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Real.cosh (f x)) (Real.sinh (f x) • f') s x
-- DifferentiableWithinAt.cosh
theorem DifferentiableWithinAt.cosh (hf : DifferentiableWithinAt ℝ f s x) :
    DifferentiableWithinAt ℝ (fun x => Real.cosh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x)
Goal: DifferentiableWithinAt ℝ (fun x => Real.cosh (f x)) s x
-- DifferentiableAt.cosh
@[simp]
theorem DifferentiableAt.cosh (hc : DifferentiableAt ℝ f x) :
    DifferentiableAt ℝ (fun x => Real.cosh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: DifferentiableAt ℝ (fun x => Real.cosh (f x)) x
-- DifferentiableOn.cosh
theorem DifferentiableOn.cosh (hc : DifferentiableOn ℝ f s) :
    DifferentiableOn ℝ (fun x => Real.cosh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hc : DifferentiableOn ℝ f s)
Goal: DifferentiableOn ℝ (fun x => Real.cosh (f x)) s
-- Differentiable.cosh
@[simp]
theorem Differentiable.cosh (hc : Differentiable ℝ f) : Differentiable ℝ fun x => Real.cosh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hc : Differentiable ℝ f)
Goal: Differentiable ℝ fun x => Real.cosh (f x)
-- fderivWithin_cosh
theorem fderivWithin_cosh (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => Real.cosh (f x)) s x = Real.sinh (f x) • fderivWithin ℝ f s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: fderivWithin ℝ (fun x => Real.cosh (f x)) s x = Real.sinh (f x) • fderivWithin ℝ f s x
-- fderiv_cosh
@[simp]
theorem fderiv_cosh (hc : DifferentiableAt ℝ f x) :
    fderiv ℝ (fun x => Real.cosh (f x)) x = Real.sinh (f x) • fderiv ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hc : DifferentiableAt ℝ f x)
Goal: fderiv ℝ (fun x => Real.cosh (f x)) x = Real.sinh (f x) • fderiv ℝ f x
-- ContDiff.cosh
theorem ContDiff.cosh {n} (h : ContDiff ℝ n f) : ContDiff ℝ n fun x => Real.cosh (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (h : ContDiff ℝ n f)
Goal: ContDiff ℝ n fun x => Real.cosh (f x)
-- ContDiffAt.cosh
theorem ContDiffAt.cosh {n} (hf : ContDiffAt ℝ n f x) :
    ContDiffAt ℝ n (fun x => Real.cosh (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {n : ℕ∞} (hf : ContDiffAt ℝ n f x)
Goal: ContDiffAt ℝ n (fun x => Real.cosh (f x)) x
-- ContDiffOn.cosh
theorem ContDiffOn.cosh {n} (hf : ContDiffOn ℝ n f s) :
    ContDiffOn ℝ n (fun x => Real.cosh (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {n : ℕ∞} (hf : ContDiffOn ℝ n f s)
Goal: ContDiffOn ℝ n (fun x => Real.cosh (f x)) s
-- ContDiffWithinAt.cosh
theorem ContDiffWithinAt.cosh {n} (hf : ContDiffWithinAt ℝ n f s x) :
    ContDiffWithinAt ℝ n (fun x => Real.cosh (f x)) s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {s : Set E} {n : ℕ∞} (hf : ContDiffWithinAt ℝ n f s x)
Goal: ContDiffWithinAt ℝ n (fun x => Real.cosh (f x)) s x
-- HasStrictFDerivAt.sinh
theorem HasStrictFDerivAt.sinh (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') x
-- HasFDerivAt.sinh
theorem HasFDerivAt.sinh (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') x
-- HasFDerivWithinAt.sinh
theorem HasFDerivWithinAt.sinh (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {f' : E →L[ℝ] ℝ} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => Real.sinh (f x)) (Real.cosh (f x) • f') s x