l3lab/ntp-mathlib huggingface dataset:
/-- **Cauchy integral formula** for the value at the center of a disc. If `f` is continuous on a
punctured closed disc of radius `R`, is differentiable at all but countably many points of the
interior of this disc, and has a limit `y` at the center of the disc, then the integral
$\oint_{‖z-c‖=R} \frac{f(z)}{z-c}\,dz$ is equal to `2πiy`. -/
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 := by
  rw [← sub_eq_zero, ← norm_le_zero_iff]
  refine' le_of_forall_le_of_dense fun ε ε0 => _
  obtain ⟨δ, δ0, hδ⟩ : ∃ δ > (0 : ℝ), ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
  exact ((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _
    (div_pos ε0 Real.two_pi_pos)
  obtain ⟨r, hr0, hrδ, hrR⟩ : ∃ r, 0 < r ∧ r ≤ δ ∧ r ≤ R :=
    ⟨min δ R, lt_min δ0 h0, min_le_left _ _, min_le_right _ _⟩
  have hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c} :=
    diff_subset_diff_right (singleton_subset_iff.2 <| mem_ball_self hr0)
  have hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c} :=
    diff_subset_diff_right (singleton_subset_iff.2 <| mem_closedBall_self hr0.le)
  have hzne : ∀ z ∈ sphere c r, z ≠ c := fun z hz =>
    ne_of_mem_of_not_mem hz fun h => hr0.ne' <| dist_self c ▸ Eq.symm h
  /- The integral `∮ z in C(c, r), f z / (z - c)` does not depend on `0 < r ≤ R` and tends to
    `2πIy` as `r → 0`. -/
  calc
    ‖(∮ z in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ =
        ‖(∮ z in C(c, r), (z - c)⁻¹ • f z) - ∮ z in C(c, r), (z - c)⁻¹ • y‖ := by
      congr 2
      · exact circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0
          hrR hs (hc.mono hsub) fun z hz => hd z ⟨hsub' hz.1, hz.2⟩
      · simp [hr0.ne']
    _ = ‖∮ z in C(c, r), (z - c)⁻¹ • (f z - y)‖ := by
      simp only [smul_sub]
      have hc' : ContinuousOn (fun z => (z - c)⁻¹) (sphere c r) :=
        (continuousOn_id.sub continuousOn_const).inv₀ fun z hz => sub_ne_zero.2 <| hzne _ hz
      rw [circleIntegral.integral_sub] <;> 

Proof state:
case hg
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : Set.Countable s
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
hc' : ContinuousOn (fun z => (z - c)⁻¹) (sphere c r)
⊢ CircleIntegrable (fun z => (z - c)⁻¹ • y) c r

Next tactic:
refine' (hc'.smul _).circleIntegrable hr0.le
/-- **Cauchy integral formula** for the value at the center of a disc. If `f` is continuous on a
punctured closed disc of radius `R`, is differentiable at all but countably many points of the
interior of this disc, and has a limit `y` at the center of the disc, then the integral
$\oint_{‖z-c‖=R} \frac{f(z)}{z-c}\,dz$ is equal to `2πiy`. -/
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 := by
  rw [← sub_eq_zero, ← norm_le_zero_iff]
  refine' le_of_forall_le_of_dense fun ε ε0 => _
  obtain ⟨δ, δ0, hδ⟩ : ∃ δ > (0 : ℝ), ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
  exact ((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _
    (div_pos ε0 Real.two_pi_pos)
  obtain ⟨r, hr0, hrδ, hrR⟩ : ∃ r, 0 < r ∧ r ≤ δ ∧ r ≤ R :=
    ⟨min δ R, lt_min δ0 h0, min_le_left _ _, min_le_right _ _⟩
  have hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c} :=
    diff_subset_diff_right (singleton_subset_iff.2 <| mem_ball_self hr0)
  have hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c} :=
    diff_subset_diff_right (singleton_subset_iff.2 <| mem_closedBall_self hr0.le)
  have hzne : ∀ z ∈ sphere c r, z ≠ c := fun z hz =>
    ne_of_mem_of_not_mem hz fun h => hr0.ne' <| dist_self c ▸ Eq.symm h
  /- The integral `∮ z in C(c, r), f z / (z - c)` does not depend on `0 < r ≤ R` and tends to
    `2πIy` as `r → 0`. -/
  calc
    ‖(∮ z in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ =
        ‖(∮ z in C(c, r), (z - c)⁻¹ • f z) - ∮ z in C(c, r), (z - c)⁻¹ • y‖ := by
      congr 2
      · exact circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0
          hrR hs (hc.mono hsub) fun z hz => hd z ⟨hsub' hz.1, hz.2⟩
      · simp [hr0.ne']
    _ = ‖∮ z in C(c, r), (z - c)⁻¹ • (f z - y)‖ := by
      simp only [smul_sub]
      have hc' : ContinuousOn (fun z => (z - c)⁻¹) (sphere c r) :=
        (continuousOn_id.sub continuousOn_const).inv₀ fun z hz => sub_ne_zero.2 <| hzne _ hz
      rw [circleIntegral.integral_sub] <;> refine' (hc'.smul _).circleIntegrable hr0.le
      · 

Proof state:
case hf
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → E
y : E
s : Set ℂ
hs : Set.Countable s
hc : ContinuousOn f (closedBall c R \ {c})
hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z
hy : Tendsto f (𝓝[≠] c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
δ : ℝ
δ0 : δ > 0
hδ : ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π)
r : ℝ
hr0 : 0 < r
hrδ : r ≤ δ
hrR : r ≤ R
hsub : closedBall c R \ ball c r ⊆ closedBall c R \ {c}
hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c}
hzne : ∀ z ∈ sphere c r, z ≠ c
hc' : ContinuousOn (fun z => (z - c)⁻¹) (sphere c r)
⊢ ContinuousOn (fun x => f x) (sphere c r)

Next tactic:
exact hc.mono <| subset_inter
          (sphere_subset_closedBall.trans <| closedBall_subset_closedBall hrR) hzne
/-- **Cauchy-Goursat theorem** for a disk: if `f : ℂ → E` is continuous on a closed disk
`{z | ‖z - c‖ ≤ R}` and is complex differentiable at all but countably many points of its interior,
then the integral $\oint_{|z-c|=R}f(z)\,dz$ equals zero. -/
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 := by
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
h0 : 0 ≤ R
f : ℂ → E
c : ℂ
s : Set ℂ
hs : Set.Countable s
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
⊢ (∮ (z : ℂ) in C(c, R), f z) = 0

Next tactic:
rcases h0.eq_or_lt with (rfl | h0)
/-- **Cauchy-Goursat theorem** for a disk: if `f : ℂ → E` is continuous on a closed disk
`{z | ‖z - c‖ ≤ R}` and is complex differentiable at all but countably many points of its interior,
then the integral $\oint_{|z-c|=R}f(z)\,dz$ equals zero. -/
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 := by
  rcases h0.eq_or_lt with (rfl | h0); · 

Proof state:
case inl
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
s : Set ℂ
hs : Set.Countable s
h0 : 0 ≤ 0
hc : ContinuousOn f (closedBall c 0)
hd : ∀ z ∈ ball c 0 \ s, DifferentiableAt ℂ f z
⊢ (∮ (z : ℂ) in C(c, 0), f z) = 0

Next tactic:
apply circleIntegral.integral_radius_zero
/-- **Cauchy-Goursat theorem** for a disk: if `f : ℂ → E` is continuous on a closed disk
`{z | ‖z - c‖ ≤ R}` and is complex differentiable at all but countably many points of its interior,
then the integral $\oint_{|z-c|=R}f(z)\,dz$ equals zero. -/
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 := by
  rcases h0.eq_or_lt with (rfl | h0); · apply circleIntegral.integral_radius_zero
  

Proof state:
case inr
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
h0✝ : 0 ≤ R
f : ℂ → E
c : ℂ
s : Set ℂ
hs : Set.Countable s
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
h0 : 0 < R
⊢ (∮ (z : ℂ) in C(c, R), f z) = 0

Next tactic:
calc
    (∮ z in C(c, R), f z) = ∮ z in C(c, R), (z - c)⁻¹ • (z - c) • f z :=
      (circleIntegral.integral_sub_inv_smul_sub_smul _ _ _ _).symm
    _ = (2 * ↑π * I : ℂ) • (c - c) • f c :=
      (circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable h0 hs
        ((continuousOn_id.sub continuousOn_const).smul hc) fun z hz =>
        (differentiableAt_id.sub_const _).smul (hd z hz))
    _ = 0 := by rw [sub_self, zero_smul, smul_zero]
/-- **Cauchy-Goursat theorem** for a disk: if `f : ℂ → E` is continuous on a closed disk
`{z | ‖z - c‖ ≤ R}` and is complex differentiable at all but countably many points of its interior,
then the integral $\oint_{|z-c|=R}f(z)\,dz$ equals zero. -/
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 := by
  rcases h0.eq_or_lt with (rfl | h0); · apply circleIntegral.integral_radius_zero
  calc
    (∮ z in C(c, R), f z) = ∮ z in C(c, R), (z - c)⁻¹ • (z - c) • f z :=
      (circleIntegral.integral_sub_inv_smul_sub_smul _ _ _ _).symm
    _ = (2 * ↑π * I : ℂ) • (c - c) • f c :=
      (circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable h0 hs
        ((continuousOn_id.sub continuousOn_const).smul hc) fun z hz =>
        (differentiableAt_id.sub_const _).smul (hd z hz))
    _ = 0 := by 

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
h0✝ : 0 ≤ R
f : ℂ → E
c : ℂ
s : Set ℂ
hs : Set.Countable s
hc : ContinuousOn f (closedBall c R)
hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z
h0 : 0 < R
⊢ (2 * ↑π * I) • (c - c) • f c = 0

Next tactic:
rw [sub_self, zero_smul, smul_zero]
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hR : 0 < R := dist_nonneg.trans_lt hw.1
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
set F : ℂ → E := dslope f w
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hws : (insert w s).Countable := hs.insert w
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
hcF : ContinuousOn F (closedBall c R)
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (diff_subset_diff_right (subset_insert _ _) hz))
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
  have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (diff_subset_diff_right (subset_insert _ _) hz))
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have HI := circleIntegral_eq_zero_of_differentiable_on_off_countable hR.le hws hcF hdF
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
  have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (diff_subset_diff_right (subset_insert _ _) hz))
  have HI := circleIntegral_eq_zero_of_differentiable_on_off_countable hR.le hws hcF hdF
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
HI : (∮ (z : ℂ) in C(c, R), F z) = 0
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hne : ∀ z ∈ sphere c R, z ≠ w := fun z hz => ne_of_mem_of_not_mem hz (ne_of_lt hw.1)
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
  have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (diff_subset_diff_right (subset_insert _ _) hz))
  have HI := circleIntegral_eq_zero_of_differentiable_on_off_countable hR.le hws hcF hdF
  have hne : ∀ z ∈ sphere c R, z ≠ w := fun z hz => ne_of_mem_of_not_mem hz (ne_of_lt hw.1)
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
HI : (∮ (z : ℂ) in C(c, R), F z) = 0
hne : ∀ z ∈ sphere c R, z ≠ w
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hFeq : EqOn F (fun z => (z - w)⁻¹ • f z - (z - w)⁻¹ • f w) (sphere c R) := fun z hz ↦
    calc
      F z = (z - w)⁻¹ • (f z - f w) := update_noteq (hne z hz) _ _
      _ = (z - w)⁻¹ • f z - (z - w)⁻¹ • f w := smul_sub _ _ _
/-- An auxiliary lemma for
`Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable`. This lemma assumes
`w ∉ s` while the main lemma drops this assumption. -/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw.1
  set F : ℂ → E := dslope f w
  have hws : (insert w s).Countable := hs.insert w
  have hcF : ContinuousOn F (closedBall c R) :=
    (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 ⟨hc, hd _ hw⟩
  have hdF : ∀ z ∈ ball (c : ℂ) R \ insert w s, DifferentiableAt ℂ F z := fun z hz =>
    (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2
      (hd _ (diff_subset_diff_right (subset_insert _ _) hz))
  have HI := circleIntegral_eq_zero_of_differentiable_on_off_countable hR.le hws hcF hdF
  have hne : ∀ z ∈ sphere c R, z ≠ w := fun z hz => ne_of_mem_of_not_mem hz (ne_of_lt hw.1)
  have hFeq : EqOn F (fun z => (z - w)⁻¹ • f z - (z - w)⁻¹ • f w) (sphere c R) := fun z hz ↦
    calc
      F z = (z - w)⁻¹ • (f z - f w) := update_noteq (hne z hz) _ _
      _ = (z - w)⁻¹ • f z - (z - w)⁻¹ • f w := smul_sub _ _ _
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : Set.Countable s
hw : w ∈ ball c R \ s
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
F : ℂ → E := dslope f w
hws : Set.Countable (insert w s)
hcF : ContinuousOn F (closedBall c R)
hdF : ∀ z ∈ ball c R \ insert w s, DifferentiableAt ℂ F z
HI : (∮ (z : ℂ) in C(c, R), F z) = 0
hne : ∀ z ∈ sphere c R, z ≠ w
hFeq : EqOn F (fun z => (z - w)⁻¹ • f z - (z - w)⁻¹ • f w) (sphere c R)
⊢ (∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w

Next tactic:
have hc' : ContinuousOn (fun z => (z - w)⁻¹) (sphere c R) :=
    (continuousOn_id.sub continuousOn_const).inv₀ fun z hz => sub_ne_zero.2 <| hne z hz