ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (s : Set ℂ) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd :
  ∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x) (Hi :
  MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
    MeasureTheory.volume)
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  ∫ (x : ℝ) in z.re..w.re,
    ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) Complex.I
-- Complex.integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real
theorem integral_boundary_rect_of_continuousOn_of_hasFDerivAt_real (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im),
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E) (z w : ℂ) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : ∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im), HasFDerivAt f (f' x) x) (Hi :
  MeasureTheory.IntegrableOn (fun z => Complex.I • (f' z) 1 - (f' z) Complex.I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
    MeasureTheory.volume)
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  ∫ (x : ℝ) in z.re..w.re,
    ∫ (y : ℝ) in z.im..w.im, Complex.I • (f' (↑x + ↑y * Complex.I)) 1 - (f' (↑x + ↑y * Complex.I)) Complex.I
-- Complex.integral_boundary_rect_of_differentiableOn_real
theorem integral_boundary_rect_of_differentiableOn_real (f : ℂ → E) (z w : ℂ)
    (Hd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hi : IntegrableOn (fun z => I • fderiv ℝ f z 1 - fderiv ℝ f z I)
      ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im,
        I • fderiv ℝ f (x + y * I) 1 - fderiv ℝ f (x + y * I) I 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (z w : ℂ) (Hd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hi :
  MeasureTheory.IntegrableOn (fun z => Complex.I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) Complex.I)
    ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) MeasureTheory.volume)
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  ∫ (x : ℝ) in z.re..w.re,
    ∫ (y : ℝ) in z.im..w.im,
      Complex.I • (fderiv ℝ f (↑x + ↑y * Complex.I)) 1 - (fderiv ℝ f (↑x + ↑y * Complex.I)) Complex.I
-- Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable
theorem integral_boundary_rect_eq_zero_of_differentiable_on_off_countable (f : ℂ → E) (z w : ℂ)
    (s : Set ℂ) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      DifferentiableAt ℂ f x) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (z w : ℂ) (s : Set ℂ) (hs : s.Countable) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd :
  ∀ x ∈ Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im) \ s, DifferentiableAt ℂ f x)
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  0
-- Complex.integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn
theorem integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn (f : ℂ → E) (z w : ℂ)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : DifferentiableOn ℂ f
      (Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im))) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (z w : ℂ) (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) (Hd : DifferentiableOn ℂ f (Set.Ioo (min z.re w.re) (max z.re w.re) ×ℂ Set.Ioo (min z.im w.im) (max z.im w.im)))
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  0
-- Complex.integral_boundary_rect_eq_zero_of_differentiableOn
theorem integral_boundary_rect_eq_zero_of_differentiableOn (f : ℂ → E) (z w : ℂ)
    (H : DifferentiableOn ℂ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → E) (z w : ℂ) (H : DifferentiableOn ℂ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
Goal: (((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +
      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -
    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =
  0
-- Complex.circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable
theorem circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable {c : ℂ}
    {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable)
    (hc : ContinuousOn f (closedBall c R \ ball c r))
    (hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z) :
    (∮ z in C(c, R), (z - c)⁻¹ • f z) = ∮ z in C(c, r), (z - c)⁻¹ • f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {c : ℂ} {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c R \ Metric.ball c r)) (hd : ∀ z ∈ (Metric.ball c R \ Metric.closedBall c r) \ s, DifferentiableAt ℂ f z)
Goal: (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ • f z
-- Complex.circleIntegral_eq_of_differentiable_on_annulus_off_countable
theorem circleIntegral_eq_of_differentiable_on_annulus_off_countable {c : ℂ} {r R : ℝ} (h0 : 0 < r)
    (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable)
    (hc : ContinuousOn f (closedBall c R \ ball c r))
    (hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z) :
    (∮ z in C(c, R), f z) = ∮ z in C(c, r), f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {c : ℂ} {r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c R \ Metric.ball c r)) (hd : ∀ z ∈ (Metric.ball c R \ Metric.closedBall c r) \ s, DifferentiableAt ℂ f z)
Goal: (∮ (z : ℂ) in C(c, R), f z) = ∮ (z : ℂ) in C(c, r), f z
-- Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto
theorem circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto {c : ℂ}
    {R : ℝ} (h0 : 0 < R) {f : ℂ → E} {y : E} {s : Set ℂ} (hs : s.Countable)
    (hc : ContinuousOn f (closedBall c R \ {c}))
    (hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z) (hy : Tendsto f (𝓝[{c}ᶜ] c) (𝓝 y)) :
    (∮ z in C(c, R), (z - c)⁻¹ • f z) = (2 * π * I : ℂ) • y 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {c : ℂ} {R : ℝ} (h0 : 0 < R) {f : ℂ → E} {y : E} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c R \ {c})) (hd : ∀ z ∈ (Metric.ball c R \ {c}) \ s, DifferentiableAt ℂ f z) (hy : Filter.Tendsto f (𝓝[≠] c) (𝓝 y))
Goal: (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * Complex.I) • y
-- Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable
theorem circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable {R : ℝ} (h0 : 0 < R)
    {f : ℂ → E} {c : ℂ} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R))
    (hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z) :
    (∮ z in C(c, R), (z - c)⁻¹ • f z) = (2 * π * I : ℂ) • f c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} (h0 : 0 < R) {f : ℂ → E} {c : ℂ} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ z ∈ Metric.ball c R \ s, DifferentiableAt ℂ f z)
Goal: (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * Complex.I) • f c
-- Complex.circleIntegral_eq_zero_of_differentiable_on_off_countable
theorem circleIntegral_eq_zero_of_differentiable_on_off_countable {R : ℝ} (h0 : 0 ≤ R) {f : ℂ → E}
    {c : ℂ} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R))
    (hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z) : (∮ z in C(c, R), f z) = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} (h0 : 0 ≤ R) {f : ℂ → E} {c : ℂ} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ z ∈ Metric.ball c R \ s, DifferentiableAt ℂ f z)
Goal: (∮ (z : ℂ) in C(c, R), f z) = 0
-- Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux
theorem circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux {R : ℝ} {c w : ℂ}
    {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R \ s)
    (hc : ContinuousOn f (closedBall c R)) (hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
    (∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ Metric.ball c R \ s) (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ x ∈ Metric.ball c R \ s, DifferentiableAt ℂ f x)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * Complex.I) • f w
-- Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
theorem two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable {R : ℝ}
    {c w : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R)
    (hc : ContinuousOn f (closedBall c R)) (hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
    ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) = f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ Metric.ball c R) (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ x ∈ Metric.ball c R \ s, DifferentiableAt ℂ f x)
Goal: ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w
-- Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
theorem circleIntegral_sub_inv_smul_of_differentiable_on_off_countable {R : ℝ} {c w : ℂ} {f : ℂ → E}
    {s : Set ℂ} (hs : s.Countable) (hw : w ∈ ball c R) (hc : ContinuousOn f (closedBall c R))
    (hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x) :
    (∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ Metric.ball c R) (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ x ∈ Metric.ball c R \ s, DifferentiableAt ℂ f x)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * Complex.I) • f w
-- DiffContOnCl.circleIntegral_sub_inv_smul
theorem _root_.DiffContOnCl.circleIntegral_sub_inv_smul {R : ℝ} {c w : ℂ} {f : ℂ → E}
    (h : DiffContOnCl ℂ f (ball c R)) (hw : w ∈ ball c R) :
    (∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} (h : DiffContOnCl ℂ f (Metric.ball c R)) (hw : w ∈ Metric.ball c R)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * Complex.I) • f w
-- DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul
theorem _root_.DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul {R : ℝ} {c w : ℂ}
    {f : ℂ → E} (hf : DiffContOnCl ℂ f (ball c R)) (hw : w ∈ ball c R) :
    ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) = f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} (hf : DiffContOnCl ℂ f (Metric.ball c R)) (hw : w ∈ Metric.ball c R)
Goal: ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w
-- DifferentiableOn.circleIntegral_sub_inv_smul
theorem _root_.DifferentiableOn.circleIntegral_sub_inv_smul {R : ℝ} {c w : ℂ} {f : ℂ → E}
    (hd : DifferentiableOn ℂ f (closedBall c R)) (hw : w ∈ ball c R) :
    (∮ z in C(c, R), (z - w)⁻¹ • f z) = (2 * π * I : ℂ) • f w 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ} {c w : ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f (Metric.closedBall c R)) (hw : w ∈ Metric.ball c R)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * Complex.I) • f w
-- Complex.circleIntegral_div_sub_of_differentiable_on_off_countable
theorem circleIntegral_div_sub_of_differentiable_on_off_countable {R : ℝ} {c w : ℂ} {s : Set ℂ}
    (hs : s.Countable) (hw : w ∈ ball c R) {f : ℂ → ℂ} (hc : ContinuousOn f (closedBall c R))
    (hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z) :
    (∮ z in C(c, R), f z / (z - w)) = 2 * π * I * f w 
Arguments: {R : ℝ} {c w : ℂ} {s : Set ℂ} (hs : s.Countable) (hw : w ∈ Metric.ball c R) {f : ℂ → ℂ} (hc : ContinuousOn f (Metric.closedBall c R)) (hd : ∀ z ∈ Metric.ball c R \ s, DifferentiableAt ℂ f z)
Goal: (∮ (z : ℂ) in C(c, R), f z / (z - w)) = 2 * ↑π * Complex.I * f w
-- Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable
theorem hasFPowerSeriesOnBall_of_differentiable_off_countable {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
    {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R))
    (hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z) (hR : 0 < R) :
    HasFPowerSeriesOnBall f (cauchyPowerSeries f c R) c R 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ≥0} {c : ℂ} {f : ℂ → E} {s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (Metric.closedBall c ↑R)) (hd : ∀ z ∈ Metric.ball c ↑R \ s, DifferentiableAt ℂ f z) (hR : 0 < R)
Goal: HasFPowerSeriesOnBall f (cauchyPowerSeries f c ↑R) c ↑R
-- DiffContOnCl.hasFPowerSeriesOnBall
theorem _root_.DiffContOnCl.hasFPowerSeriesOnBall {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
    (hf : DiffContOnCl ℂ f (ball c R)) (hR : 0 < R) :
    HasFPowerSeriesOnBall f (cauchyPowerSeries f c R) c R 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ≥0} {c : ℂ} {f : ℂ → E} (hf : DiffContOnCl ℂ f (Metric.ball c ↑R)) (hR : 0 < R)
Goal: HasFPowerSeriesOnBall f (cauchyPowerSeries f c ↑R) c ↑R
-- DifferentiableOn.hasFPowerSeriesOnBall
protected theorem _root_.DifferentiableOn.hasFPowerSeriesOnBall {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
    (hd : DifferentiableOn ℂ f (closedBall c R)) (hR : 0 < R) :
    HasFPowerSeriesOnBall f (cauchyPowerSeries f c R) c R 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {R : ℝ≥0} {c : ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f (Metric.closedBall c ↑R)) (hR : 0 < R)
Goal: HasFPowerSeriesOnBall f (cauchyPowerSeries f c ↑R) c ↑R
-- DifferentiableOn.analyticAt
protected theorem _root_.DifferentiableOn.analyticAt {s : Set ℂ} {f : ℂ → E} {z : ℂ}
    (hd : DifferentiableOn ℂ f s) (hz : s ∈ 𝓝 z) : AnalyticAt ℂ f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E} {z : ℂ} (hd : DifferentiableOn ℂ f s) (hz : s ∈ 𝓝 z)
Goal: AnalyticAt ℂ f z
-- DifferentiableOn.analyticOn
theorem _root_.DifferentiableOn.analyticOn {s : Set ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f s)
    (hs : IsOpen s) : AnalyticOn ℂ f s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f s) (hs : IsOpen s)
Goal: AnalyticOn ℂ f s
-- DifferentiableOn.contDiffOn
protected theorem _root_.DifferentiableOn.contDiffOn {s : Set ℂ} {f : ℂ → E} {n : ℕ}
    (hd : DifferentiableOn ℂ f s) (hs : IsOpen s) : ContDiffOn ℂ n f s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {s : Set ℂ} {f : ℂ → E} {n : ℕ} (hd : DifferentiableOn ℂ f s) (hs : IsOpen s)
Goal: ContDiffOn ℂ (↑n) f s
-- Differentiable.analyticAt
protected theorem _root_.Differentiable.analyticAt {f : ℂ → E} (hf : Differentiable ℂ f) (z : ℂ) :
    AnalyticAt ℂ f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} (hf : Differentiable ℂ f) (z : ℂ)
Goal: AnalyticAt ℂ f z
-- Differentiable.contDiff
protected theorem _root_.Differentiable.contDiff {f : ℂ → E} (hf : Differentiable ℂ f) {n : ℕ∞} :
    ContDiff ℂ n f 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} (hf : Differentiable ℂ f) {n : ℕ∞}
Goal: ContDiff ℂ n f
-- Differentiable.hasFPowerSeriesOnBall
protected theorem _root_.Differentiable.hasFPowerSeriesOnBall {f : ℂ → E} (h : Differentiable ℂ f)
    (z : ℂ) {R : ℝ≥0} (hR : 0 < R) : HasFPowerSeriesOnBall f (cauchyPowerSeries f z R) z ∞ 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} (h : Differentiable ℂ f) (z : ℂ) {R : ℝ≥0} (hR : 0 < R)
Goal: HasFPowerSeriesOnBall f (cauchyPowerSeries f z ↑R) z ⊤
-- Complex.analyticOn_iff_differentiableOn
theorem analyticOn_iff_differentiableOn {f : ℂ → E} {s : Set ℂ} (o : IsOpen s) :
    AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {s : Set ℂ} (o : IsOpen s)
Goal: AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s
-- Complex.analyticOn_univ_iff_differentiable
theorem analyticOn_univ_iff_differentiable {f : ℂ → E} :
    AnalyticOn ℂ f univ ↔ Differentiable ℂ f 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E}
Goal: AnalyticOn ℂ f Set.univ ↔ Differentiable ℂ f
-- Complex.analyticAt_iff_eventually_differentiableAt
theorem analyticAt_iff_eventually_differentiableAt {f : ℂ → E} {c : ℂ} :
    AnalyticAt ℂ f c ↔ ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E} {c : ℂ}
Goal: AnalyticAt ℂ f c ↔ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
-- SSet.KanComplex.hornFilling
class KanComplex (S : SSet) : Prop where
  hornFilling : ∀ ⦃n : ℕ⦄ ⦃i : Fin (n+1)⦄ (σ₀ : Λ[n, i] ⟶ S),
    ∃ σ : Δ[n] ⟶ S, σ₀ = hornInclusion n i ≫ σ
Arguments: {S : SSet} [self : S.KanComplex] ⦃n : ℕ⦄ ⦃i : Fin (n + 1)⦄ (σ₀ : SSet.horn n i ⟶ S)
Goal: ∃ σ, σ₀ = CategoryTheory.CategoryStruct.comp (SSet.hornInclusion n i) σ
-- SSet.hom_ext
@[ext]
lemma hom_ext {X Y : SSet} {f g : X ⟶ Y} (w : ∀ n, f.app n = g.app n) : f = g 
Arguments: {X Y : SSet} {f g : X ⟶ Y} (w : ∀ (n : SimplexCategoryᵒᵖ), f.app n = g.app n)
Goal: f = g
-- SSet.standardSimplex.map_id
@[simp]
lemma map_id (n : SimplexCategory) :
    (SSet.standardSimplex.map (SimplexCategory.Hom.mk OrderHom.id : n ⟶ n)) = 𝟙 _ 
Arguments: (n : SimplexCategory)
Goal: SSet.standardSimplex.map (SimplexCategory.Hom.mk OrderHom.id) = 𝟙 (SSet.standardSimplex.obj n)
-- SSet.standardSimplex.map_apply
lemma map_apply {m₁ m₂ : SimplexCategoryᵒᵖ} (f : m₁ ⟶ m₂) {n : SimplexCategory}
    (x : (standardSimplex.{u}.obj n).obj m₁) :
    (standardSimplex.{u}.obj n).map f x = (objEquiv _ _).symm (f.unop ≫ (objEquiv _ _) x) 
Arguments: {m₁ m₂ : SimplexCategoryᵒᵖ} (f : m₁ ⟶ m₂) {n : SimplexCategory} (x : (SSet.standardSimplex.obj n).obj m₁)
Goal: (SSet.standardSimplex.obj n).map f x =
  (SSet.standardSimplex.objEquiv n m₂).symm (f.unop ≫ (SSet.standardSimplex.objEquiv n m₁) x)
-- SSet.standardSimplex.const_down_toOrderHom
@[simp]
lemma const_down_toOrderHom (n : ℕ) (k : Fin (n+1)) (m : SimplexCategoryᵒᵖ) :
    (const n k m).down.toOrderHom = OrderHom.const _ k 
Arguments: (n : ℕ) (k : Fin (n + 1)) (m : SimplexCategoryᵒᵖ)
Goal: SimplexCategory.Hom.toOrderHom (SSet.standardSimplex.const n k m).down =
  (OrderHom.const (Fin ((Opposite.unop m).len + 1))) k
-- SSet.standardSimplex.coe_edge_down_toOrderHom
lemma coe_edge_down_toOrderHom (n : ℕ) (a b : Fin (n+1)) (hab : a ≤ b) :
    ↑(edge n a b hab).down.toOrderHom = ![a, b] 
Arguments: (n : ℕ) (a b : Fin (n + 1)) (hab : a ≤ b)
Goal: ⇑(SimplexCategory.Hom.toOrderHom (SSet.standardSimplex.edge n a b hab).down) = ![a, b]
-- SSet.standardSimplex.coe_triangle_down_toOrderHom
lemma coe_triangle_down_toOrderHom {n : ℕ} (a b c : Fin (n+1)) (hab : a ≤ b) (hbc : b ≤ c) :
    ↑(triangle a b c hab hbc).down.toOrderHom = ![a, b, c] 
Arguments: {n : ℕ} (a b c : Fin (n + 1)) (hab : a ≤ b) (hbc : b ≤ c)
Goal: ⇑(SimplexCategory.Hom.toOrderHom (SSet.standardSimplex.triangle a b c hab hbc).down) = ![a, b, c]
-- SSet.horn.const_coe
@[simps]
def const (n : ℕ) (i k : Fin (n+3)) (m : SimplexCategoryᵒᵖ) : Λ[n+2, i].obj m 
Arguments: (n : ℕ) (i k : Fin (n + 3)) (m : SimplexCategoryᵒᵖ)
Goal: ↑(SSet.horn.const n i k m) = SSet.standardSimplex.const (n + 2) k m
-- SSet.horn.edge_coe
@[simps]
def edge (n : ℕ) (i a b : Fin (n+1)) (hab : a ≤ b) (H : Finset.card {i, a, b} ≤ n) :
    Λ[n, i] _[1] 
Arguments: (n : ℕ) (i a b : Fin (n + 1)) (hab : a ≤ b) (H : {i, a, b}.card ≤ n)
Goal: ↑(SSet.horn.edge n i a b hab H) = SSet.standardSimplex.edge n a b hab
-- SSet.horn.edge₃_coe_down
@[simps!]
def edge₃ (n : ℕ) (i a b : Fin (n+1)) (hab : a ≤ b) (H : 3 ≤ n) :
    Λ[n, i] _[1] 
Arguments: (n : ℕ) (i a b : Fin (n + 1)) (hab : a ≤ b) (H : 3 ≤ n)
Goal: (↑(SSet.horn.edge₃ n i a b hab H)).down = SimplexCategory.Hom.mk { toFun := ![a, b], monotone' := ⋯ }