ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- rexp_neg_quadratic_isLittleO_rpow_atTop
lemma rexp_neg_quadratic_isLittleO_rpow_atTop {a : ℝ} (ha : a < 0) (b s : ℝ) :
    (fun x ↦ rexp (a * x ^ 2 + b * x)) =o[atTop] (· ^ s) 
Arguments: {a : ℝ} (ha : a < 0) (b s : ℝ)
Goal: (fun x => rexp (a * x ^ 2 + b * x)) =o[Filter.atTop] fun x => x ^ s
-- cexp_neg_quadratic_isLittleO_rpow_atTop
lemma cexp_neg_quadratic_isLittleO_rpow_atTop {a : ℂ} (ha : a.re < 0) (b : ℂ) (s : ℝ) :
    (fun x : ℝ ↦ cexp (a * x ^ 2 + b * x)) =o[atTop] (· ^ s) 
Arguments: {a : ℂ} (ha : a.re < 0) (b : ℂ) (s : ℝ)
Goal: (fun x => cexp (a * ↑x ^ 2 + b * ↑x)) =o[Filter.atTop] fun x => x ^ s
-- cexp_neg_quadratic_isLittleO_abs_rpow_cocompact
lemma cexp_neg_quadratic_isLittleO_abs_rpow_cocompact {a : ℂ} (ha : a.re < 0) (b : ℂ) (s : ℝ) :
    (fun x : ℝ ↦ cexp (a * x ^ 2 + b * x)) =o[cocompact ℝ] (|·| ^ s) 
Arguments: {a : ℂ} (ha : a.re < 0) (b : ℂ) (s : ℝ)
Goal: (fun x => cexp (a * ↑x ^ 2 + b * ↑x)) =o[Filter.cocompact ℝ] fun x => |x| ^ s
-- tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact
theorem tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact {a : ℝ} (ha : 0 < a) (s : ℝ) :
    Tendsto (fun x : ℝ => |x| ^ s * rexp (-a * x ^ 2)) (cocompact ℝ) (𝓝 0) 
Arguments: {a : ℝ} (ha : 0 < a) (s : ℝ)
Goal: Filter.Tendsto (fun x => |x| ^ s * rexp (-a * x ^ 2)) (Filter.cocompact ℝ) (𝓝 0)
-- isLittleO_exp_neg_mul_sq_cocompact
theorem isLittleO_exp_neg_mul_sq_cocompact {a : ℂ} (ha : 0 < a.re) (s : ℝ) :
    (fun x : ℝ => Complex.exp (-a * x ^ 2)) =o[cocompact ℝ] fun x : ℝ => |x| ^ s 
Arguments: {a : ℂ} (ha : 0 < a.re) (s : ℝ)
Goal: (fun x => cexp (-a * ↑x ^ 2)) =o[Filter.cocompact ℝ] fun x => |x| ^ s
-- Complex.tsum_exp_neg_quadratic
theorem Complex.tsum_exp_neg_quadratic {a : ℂ} (ha : 0 < a.re) (b : ℂ) :
    (∑' n : ℤ, cexp (-π * a * n ^ 2 + 2 * π * b * n)) =
      1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n + I * b) ^ 2) 
Arguments: {a : ℂ} (ha : 0 < a.re) (b : ℂ)
Goal: ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2 + 2 * ↑π * b * ↑n) =
  1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * (↑n + Complex.I * b) ^ 2)
-- Complex.tsum_exp_neg_mul_int_sq
theorem Complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
    (∑' n : ℤ, cexp (-π * a * (n : ℂ) ^ 2)) =
      1 / a ^ (1 / 2 : ℂ) * ∑' n : ℤ, cexp (-π / a * (n : ℂ) ^ 2) 
Arguments: {a : ℂ} (ha : 0 < a.re)
Goal: ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)
-- Real.tsum_exp_neg_mul_int_sq
theorem Real.tsum_exp_neg_mul_int_sq {a : ℝ} (ha : 0 < a) :
    (∑' n : ℤ, exp (-π * a * (n : ℝ) ^ 2)) =
      (1 : ℝ) / a ^ (1 / 2 : ℝ) * (∑' n : ℤ, exp (-π / a * (n : ℝ) ^ 2)) 
Arguments: {a : ℝ} (ha : 0 < a)
Goal: ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)
-- Real.hasStrictDerivAt_log_of_pos
theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x 
Arguments: {x : ℝ} (hx : 0 < x)
Goal: HasStrictDerivAt Real.log x⁻¹ x
-- Real.hasStrictDerivAt_log
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: HasStrictDerivAt Real.log x⁻¹ x
-- Real.hasDerivAt_log
theorem hasDerivAt_log (hx : x ≠ 0) : HasDerivAt log x⁻¹ x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: HasDerivAt Real.log x⁻¹ x
-- Real.differentiableAt_log
theorem differentiableAt_log (hx : x ≠ 0) : DifferentiableAt ℝ log x 
Arguments: {x : ℝ} (hx : x ≠ 0)
Goal: DifferentiableAt ℝ Real.log x
-- Real.differentiableOn_log
theorem differentiableOn_log : DifferentiableOn ℝ log {0}ᶜ 
Goal: DifferentiableOn ℝ Real.log {0}ᶜ
-- Real.differentiableAt_log_iff
@[simp]
theorem differentiableAt_log_iff : DifferentiableAt ℝ log x ↔ x ≠ 0 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.log x ↔ x ≠ 0
-- Real.deriv_log
theorem deriv_log (x : ℝ) : deriv log x = x⁻¹ 
Arguments: (x : ℝ)
Goal: deriv Real.log x = x⁻¹
-- Real.deriv_log'
@[simp]
theorem deriv_log' : deriv log = Inv.inv 
Goal: deriv Real.log = Inv.inv
-- Real.contDiffOn_log
theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ 
Arguments: {n : ℕ∞}
Goal: ContDiffOn ℝ n Real.log {0}ᶜ
-- Real.contDiffAt_log
theorem contDiffAt_log {n : ℕ∞} : ContDiffAt ℝ n log x ↔ x ≠ 0 
Arguments: {x : ℝ} {n : ℕ∞}
Goal: ContDiffAt ℝ n Real.log x ↔ x ≠ 0
-- HasDerivWithinAt.log
theorem HasDerivWithinAt.log (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0) :
    HasDerivWithinAt (fun y => log (f y)) (f' / f x) s x 
Arguments: {f : ℝ → ℝ} {x f' : ℝ} {s : Set ℝ} (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0)
Goal: HasDerivWithinAt (fun y => Real.log (f y)) (f' / f x) s x
-- HasDerivAt.log
theorem HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) :
    HasDerivAt (fun y => log (f y)) (f' / f x) x 
Arguments: {f : ℝ → ℝ} {x f' : ℝ} (hf : HasDerivAt f f' x) (hx : f x ≠ 0)
Goal: HasDerivAt (fun y => Real.log (f y)) (f' / f x) x
-- HasStrictDerivAt.log
theorem HasStrictDerivAt.log (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) :
    HasStrictDerivAt (fun y => log (f y)) (f' / f x) x 
Arguments: {f : ℝ → ℝ} {x f' : ℝ} (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0)
Goal: HasStrictDerivAt (fun y => Real.log (f y)) (f' / f x) x
-- derivWithin.log
theorem derivWithin.log (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0)
    (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => log (f x)) s x = derivWithin f s x / f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} {s : Set ℝ} (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: derivWithin (fun x => Real.log (f x)) s x = derivWithin f s x / f x
-- deriv.log
@[simp]
theorem deriv.log (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) :
    deriv (fun x => log (f x)) x = deriv f x / f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0)
Goal: deriv (fun x => Real.log (f x)) x = deriv f x / f x
-- Real.deriv_log_comp_eq_logDeriv
lemma Real.deriv_log_comp_eq_logDeriv {f : ℝ → ℝ} {x : ℝ} (h₁ : DifferentiableAt ℝ f x)
    (h₂ : f x ≠ 0) : deriv (log ∘ f) x = logDeriv f x 
Arguments: {f : ℝ → ℝ} {x : ℝ} (h₁ : DifferentiableAt ℝ f x) (h₂ : f x ≠ 0)
Goal: deriv (Real.log ∘ f) x = logDeriv f x
-- HasFDerivWithinAt.log
theorem HasFDerivWithinAt.log (hf : HasFDerivWithinAt f f' s x) (hx : f x ≠ 0) :
    HasFDerivWithinAt (fun x => log (f x)) ((f x)⁻¹ • f') s x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {f' : E →L[ℝ] ℝ} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hx : f x ≠ 0)
Goal: HasFDerivWithinAt (fun x => Real.log (f x)) ((f x)⁻¹ • f') s x
-- HasFDerivAt.log
theorem HasFDerivAt.log (hf : HasFDerivAt f f' x) (hx : f x ≠ 0) :
    HasFDerivAt (fun x => log (f x)) ((f x)⁻¹ • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {f' : E →L[ℝ] ℝ} (hf : HasFDerivAt f f' x) (hx : f x ≠ 0)
Goal: HasFDerivAt (fun x => Real.log (f x)) ((f x)⁻¹ • f') x
-- HasStrictFDerivAt.log
theorem HasStrictFDerivAt.log (hf : HasStrictFDerivAt f f' x) (hx : f x ≠ 0) :
    HasStrictFDerivAt (fun x => log (f x)) ((f x)⁻¹ • f') x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} {f' : E →L[ℝ] ℝ} (hf : HasStrictFDerivAt f f' x) (hx : f x ≠ 0)
Goal: HasStrictFDerivAt (fun x => Real.log (f x)) ((f x)⁻¹ • f') x
-- DifferentiableWithinAt.log
theorem DifferentiableWithinAt.log (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0) :
    DifferentiableWithinAt ℝ (fun x => log (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) (hx : f x ≠ 0)
Goal: DifferentiableWithinAt ℝ (fun x => Real.log (f x)) s x
-- DifferentiableAt.log
@[simp]
theorem DifferentiableAt.log (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) :
    DifferentiableAt ℝ (fun x => log (f x)) x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0)
Goal: DifferentiableAt ℝ (fun x => Real.log (f x)) x
-- ContDiffAt.log
theorem ContDiffAt.log {n} (hf : ContDiffAt ℝ n f x) (hx : f x ≠ 0) :
    ContDiffAt ℝ n (fun x => log (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) (hx : f x ≠ 0)
Goal: ContDiffAt ℝ n (fun x => Real.log (f x)) x
-- ContDiffWithinAt.log
theorem ContDiffWithinAt.log {n} (hf : ContDiffWithinAt ℝ n f s x) (hx : f x ≠ 0) :
    ContDiffWithinAt ℝ n (fun x => log (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) (hx : f x ≠ 0)
Goal: ContDiffWithinAt ℝ n (fun x => Real.log (f x)) s x
-- ContDiffOn.log
theorem ContDiffOn.log {n} (hf : ContDiffOn ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0) :
    ContDiffOn ℝ n (fun x => log (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) (hs : ∀ x ∈ s, f x ≠ 0)
Goal: ContDiffOn ℝ n (fun x => Real.log (f x)) s
-- ContDiff.log
theorem ContDiff.log {n} (hf : ContDiff ℝ n f) (h : ∀ x, f x ≠ 0) :
    ContDiff ℝ n fun x => log (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {n : ℕ∞} (hf : ContDiff ℝ n f) (h : ∀ (x : E), f x ≠ 0)
Goal: ContDiff ℝ n fun x => Real.log (f x)
-- DifferentiableOn.log
theorem DifferentiableOn.log (hf : DifferentiableOn ℝ f s) (hx : ∀ x ∈ s, f x ≠ 0) :
    DifferentiableOn ℝ (fun x => log (f x)) s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} (hf : DifferentiableOn ℝ f s) (hx : ∀ x ∈ s, f x ≠ 0)
Goal: DifferentiableOn ℝ (fun x => Real.log (f x)) s
-- Differentiable.log
@[simp]
theorem Differentiable.log (hf : Differentiable ℝ f) (hx : ∀ x, f x ≠ 0) :
    Differentiable ℝ fun x => log (f x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} (hf : Differentiable ℝ f) (hx : ∀ (x : E), f x ≠ 0)
Goal: Differentiable ℝ fun x => Real.log (f x)
-- fderivWithin.log
theorem fderivWithin.log (hf : DifferentiableWithinAt ℝ f s x) (hx : f x ≠ 0)
    (hxs : UniqueDiffWithinAt ℝ s x) :
    fderivWithin ℝ (fun x => log (f x)) s x = (f x)⁻¹ • 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) (hx : f x ≠ 0) (hxs : UniqueDiffWithinAt ℝ s x)
Goal: fderivWithin ℝ (fun x => Real.log (f x)) s x = (f x)⁻¹ • fderivWithin ℝ f s x
-- fderiv.log
@[simp]
theorem fderiv.log (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0) :
    fderiv ℝ (fun x => log (f x)) x = (f x)⁻¹ • fderiv ℝ f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0)
Goal: fderiv ℝ (fun x => Real.log (f x)) x = (f x)⁻¹ • fderiv ℝ f x
-- Real.tendsto_mul_log_one_plus_div_atTop
theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
    Tendsto (fun x => x * log (1 + t / x)) atTop (𝓝 t) 
Arguments: (t : ℝ)
Goal: Filter.Tendsto (fun x => x * Real.log (1 + t / x)) Filter.atTop (𝓝 t)
-- Real.abs_log_sub_add_sum_range_le
theorem abs_log_sub_add_sum_range_le {x : ℝ} (h : |x| < 1) (n : ℕ) :
    |(∑ i ∈ range n, x ^ (i + 1) / (i + 1)) + log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|) 
Arguments: {x : ℝ} (h : |x| < 1) (n : ℕ)
Goal: |∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + Real.log (1 - x)| ≤ |x| ^ (n + 1) / (1 - |x|)
-- Real.hasSum_pow_div_log_of_abs_lt_one
theorem hasSum_pow_div_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
    HasSum (fun n : ℕ => x ^ (n + 1) / (n + 1)) (-log (1 - x)) 
Arguments: {x : ℝ} (h : |x| < 1)
Goal: HasSum (fun n => x ^ (n + 1) / (↑n + 1)) (-Real.log (1 - x))
-- Real.hasSum_log_sub_log_of_abs_lt_one
theorem hasSum_log_sub_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
    HasSum (fun k : ℕ => (2 : ℝ) * (1 / (2 * k + 1)) * x ^ (2 * k + 1))
      (log (1 + x) - log (1 - x)) 
Arguments: {x : ℝ} (h : |x| < 1)
Goal: HasSum (fun k => 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)) (Real.log (1 + x) - Real.log (1 - x))
-- Real.hasSum_log_one_add_inv
theorem hasSum_log_one_add_inv {a : ℝ} (h : 0 < a) :
    HasSum (fun k : ℕ => (2 : ℝ) * (1 / (2 * k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1))
      (log (1 + a⁻¹)) 
Arguments: {a : ℝ} (h : 0 < a)
Goal: HasSum (fun k => 2 * (1 / (2 * ↑k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1)) (Real.log (1 + a⁻¹))
-- not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
    [CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [CompleteSpace E] {f : ℝ → E} {g : ℝ → F} {k : Set ℝ} (l : Filter ℝ) [l.NeBot] [Filter.TendstoIxxClass Set.Icc l l] (hl : k ∈ l) (hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x) (hf : Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop) (hfg : deriv f =O[l] g)
Goal: ¬MeasureTheory.IntegrableOn g k MeasureTheory.volume
-- not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter
    {f : ℝ → E} {g : ℝ → F}
    {k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
    (hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntegrableOn g k 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] {f : ℝ → E} {g : ℝ → F} {k : Set ℝ} (l : Filter ℝ) [l.NeBot] [Filter.TendstoIxxClass Set.Icc l l] (hl : k ∈ l) (hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x) (hf : Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop) (hfg : deriv f =O[l] g)
Goal: ¬MeasureTheory.IntegrableOn g k MeasureTheory.volume
-- not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter {f : ℝ → E} {g : ℝ → F}
    {a b : ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l] (hl : [[a, b]] ∈ l)
    (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
    (hfg : deriv f =O[l] g) : ¬IntervalIntegrable g volume a b 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] {f : ℝ → E} {g : ℝ → F} {a b : ℝ} (l : Filter ℝ) [l.NeBot] [Filter.TendstoIxxClass Set.Icc l l] (hl : [[a, b]] ∈ l) (hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x) (hf : Filter.Tendsto (fun x => ‖f x‖) l Filter.atTop) (hfg : deriv f =O[l] g)
Goal: ¬IntervalIntegrable g MeasureTheory.volume a b
-- not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton
    {f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (hne : a ≠ b) (hc : c ∈ [[a, b]])
    (h_deriv : ∀ᶠ x in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x)
    (h_infty : Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) atTop)
    (hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g) : ¬IntervalIntegrable g volume a b 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] {f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (hne : a ≠ b) (hc : c ∈ [[a, b]]) (h_deriv : ∀ᶠ (x : ℝ) in 𝓝[[[a, b]] \ {c}] c, DifferentiableAt ℝ f x) (h_infty : Filter.Tendsto (fun x => ‖f x‖) (𝓝[[[a, b]] \ {c}] c) Filter.atTop) (hg : deriv f =O[𝓝[[[a, b]] \ {c}] c] g)
Goal: ¬IntervalIntegrable g MeasureTheory.volume a b
-- not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured
theorem not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_punctured {f : ℝ → E}
    {g : ℝ → F} {a b c : ℝ} (h_deriv : ∀ᶠ x in 𝓝[≠] c, DifferentiableAt ℝ f x)
    (h_infty : Tendsto (fun x => ‖f x‖) (𝓝[≠] c) atTop) (hg : deriv f =O[𝓝[≠] c] g) (hne : a ≠ b)
    (hc : c ∈ [[a, b]]) : ¬IntervalIntegrable g volume a b 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] {f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (h_deriv : ∀ᶠ (x : ℝ) in 𝓝[≠] c, DifferentiableAt ℝ f x) (h_infty : Filter.Tendsto (fun x => ‖f x‖) (𝓝[≠] c) Filter.atTop) (hg : deriv f =O[𝓝[≠] c] g) (hne : a ≠ b) (hc : c ∈ [[a, b]])
Goal: ¬IntervalIntegrable g MeasureTheory.volume a b
-- not_intervalIntegrable_of_sub_inv_isBigO_punctured
theorem not_intervalIntegrable_of_sub_inv_isBigO_punctured {f : ℝ → F} {a b c : ℝ}
    (hf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f) (hne : a ≠ b) (hc : c ∈ [[a, b]]) :
    ¬IntervalIntegrable f volume a b 
Arguments: {F : Type u_2} [NormedAddCommGroup F] {f : ℝ → F} {a b c : ℝ} (hf : (fun x => (x - c)⁻¹) =O[𝓝[≠] c] f) (hne : a ≠ b) (hc : c ∈ [[a, b]])
Goal: ¬IntervalIntegrable f MeasureTheory.volume a b
-- intervalIntegrable_sub_inv_iff
@[simp]
theorem intervalIntegrable_sub_inv_iff {a b c : ℝ} :
    IntervalIntegrable (fun x => (x - c)⁻¹) volume a b ↔ a = b ∨ c ∉ [[a, b]] 
Arguments: {a b c : ℝ}
Goal: IntervalIntegrable (fun x => (x - c)⁻¹) MeasureTheory.volume a b ↔ a = b ∨ c ∉ [[a, b]]
-- intervalIntegrable_inv_iff
@[simp]
theorem intervalIntegrable_inv_iff {a b : ℝ} :
    IntervalIntegrable (fun x => x⁻¹) volume a b ↔ a = b ∨ (0 : ℝ) ∉ [[a, b]] 
Arguments: {a b : ℝ}
Goal: IntervalIntegrable (fun x => x⁻¹) MeasureTheory.volume a b ↔ a = b ∨ 0 ∉ [[a, b]]
-- not_IntegrableOn_Ici_inv
theorem not_IntegrableOn_Ici_inv {a : ℝ} :
    ¬ IntegrableOn (fun x => x⁻¹) (Ici a) 
Arguments: {a : ℝ}
Goal: ¬MeasureTheory.IntegrableOn (fun x => x⁻¹) (Set.Ici a) MeasureTheory.volume
-- not_IntegrableOn_Ioi_inv
theorem not_IntegrableOn_Ioi_inv {a : ℝ} :
    ¬ IntegrableOn (·⁻¹) (Ioi a) 
Arguments: {a : ℝ}
Goal: ¬MeasureTheory.IntegrableOn (fun x => x⁻¹) (Set.Ioi a) MeasureTheory.volume
-- ConvexOn.dual
theorem ConvexOn.dual (hf : ConvexOn 𝕜 s f) : ConcaveOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConvexOn 𝕜 s f)
Goal: ConcaveOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- ConcaveOn.dual
theorem ConcaveOn.dual (hf : ConcaveOn 𝕜 s f) : ConvexOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : ConcaveOn 𝕜 s f)
Goal: ConvexOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- StrictConvexOn.dual
theorem StrictConvexOn.dual (hf : StrictConvexOn 𝕜 s f) : StrictConcaveOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConvexOn 𝕜 s f)
Goal: StrictConcaveOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- StrictConcaveOn.dual
theorem StrictConcaveOn.dual (hf : StrictConcaveOn 𝕜 s f) : StrictConvexOn 𝕜 s (toDual ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} (hf : StrictConcaveOn 𝕜 s f)
Goal: StrictConvexOn 𝕜 s (⇑OrderDual.toDual ∘ f)
-- convexOn_id
theorem convexOn_id {s : Set β} (hs : Convex 𝕜 s) : ConvexOn 𝕜 s _root_.id 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid β] [SMul 𝕜 β] {s : Set β} (hs : Convex 𝕜 s)
Goal: ConvexOn 𝕜 s id
-- concaveOn_id
theorem concaveOn_id {s : Set β} (hs : Convex 𝕜 s) : ConcaveOn 𝕜 s _root_.id 
Arguments: {𝕜 : Type u_1} {β : Type u_5} [OrderedSemiring 𝕜] [OrderedAddCommMonoid β] [SMul 𝕜 β] {s : Set β} (hs : Convex 𝕜 s)
Goal: ConcaveOn 𝕜 s id
-- ConvexOn.congr
theorem ConvexOn.congr (hf : ConvexOn 𝕜 s f) (hfg : EqOn f g s) : ConvexOn 𝕜 s g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConvexOn 𝕜 s f) (hfg : Set.EqOn f g s)
Goal: ConvexOn 𝕜 s g
-- ConcaveOn.congr
theorem ConcaveOn.congr (hf : ConcaveOn 𝕜 s f) (hfg : EqOn f g s) : ConcaveOn 𝕜 s g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : ConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s)
Goal: ConcaveOn 𝕜 s g
-- StrictConvexOn.congr
theorem StrictConvexOn.congr (hf : StrictConvexOn 𝕜 s f) (hfg : EqOn f g s) :
    StrictConvexOn 𝕜 s g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConvexOn 𝕜 s f) (hfg : Set.EqOn f g s)
Goal: StrictConvexOn 𝕜 s g
-- StrictConcaveOn.congr
theorem StrictConcaveOn.congr (hf : StrictConcaveOn 𝕜 s f) (hfg : EqOn f g s) :
    StrictConcaveOn 𝕜 s g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f g : E → β} (hf : StrictConcaveOn 𝕜 s f) (hfg : Set.EqOn f g s)
Goal: StrictConcaveOn 𝕜 s g
-- ConvexOn.subset
theorem ConvexOn.subset {t : Set E} (hf : ConvexOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s) :
    ConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} {t : Set E} (hf : ConvexOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s)
Goal: ConvexOn 𝕜 s f
-- ConcaveOn.subset
theorem ConcaveOn.subset {t : Set E} (hf : ConcaveOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s) :
    ConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} {t : Set E} (hf : ConcaveOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s)
Goal: ConcaveOn 𝕜 s f
-- StrictConvexOn.subset
theorem StrictConvexOn.subset {t : Set E} (hf : StrictConvexOn 𝕜 t f) (hst : s ⊆ t)
    (hs : Convex 𝕜 s) : StrictConvexOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} {t : Set E} (hf : StrictConvexOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s)
Goal: StrictConvexOn 𝕜 s f
-- StrictConcaveOn.subset
theorem StrictConcaveOn.subset {t : Set E} (hf : StrictConcaveOn 𝕜 t f) (hst : s ⊆ t)
    (hs : Convex 𝕜 s) : StrictConcaveOn 𝕜 s f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 β] {s : Set E} {f : E → β} {t : Set E} (hf : StrictConcaveOn 𝕜 t f) (hst : s ⊆ t) (hs : Convex 𝕜 s)
Goal: StrictConcaveOn 𝕜 s f
-- ConvexOn.comp
theorem ConvexOn.comp (hg : ConvexOn 𝕜 (f '' s) g) (hf : ConvexOn 𝕜 s f)
    (hg' : MonotoneOn g (f '' s)) : ConvexOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : ConvexOn 𝕜 (f '' s) g) (hf : ConvexOn 𝕜 s f) (hg' : MonotoneOn g (f '' s))
Goal: ConvexOn 𝕜 s (g ∘ f)
-- ConcaveOn.comp
theorem ConcaveOn.comp (hg : ConcaveOn 𝕜 (f '' s) g) (hf : ConcaveOn 𝕜 s f)
    (hg' : MonotoneOn g (f '' s)) : ConcaveOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : ConcaveOn 𝕜 (f '' s) g) (hf : ConcaveOn 𝕜 s f) (hg' : MonotoneOn g (f '' s))
Goal: ConcaveOn 𝕜 s (g ∘ f)
-- ConvexOn.comp_concaveOn
theorem ConvexOn.comp_concaveOn (hg : ConvexOn 𝕜 (f '' s) g) (hf : ConcaveOn 𝕜 s f)
    (hg' : AntitoneOn g (f '' s)) : ConvexOn 𝕜 s (g ∘ f) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {α : Type u_4} {β : Type u_5} [OrderedSemiring 𝕜] [AddCommMonoid E] [OrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMul 𝕜 E] [SMul 𝕜 α] [SMul 𝕜 β] {s : Set E} {f : E → β} {g : β → α} (hg : ConvexOn 𝕜 (f '' s) g) (hf : ConcaveOn 𝕜 s f) (hg' : AntitoneOn g (f '' s))
Goal: ConvexOn 𝕜 s (g ∘ f)