l3lab/ntp-mathlib huggingface dataset:
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
⊢ Set.Nonempty (t ∩ (ball c R \ s))

Next tactic:
set g : ℝ → ℂ := fun x => w + ofReal x
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
⊢ Set.Nonempty (t ∩ (ball c R \ s))

Next tactic:
have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this : Tendsto g (𝓝 0) (𝓝 w)
⊢ Set.Nonempty (t ∩ (ball c R \ s))

Next tactic:
rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  

Proof state:
case intro.intro.intro
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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
⊢ Set.Nonempty (t ∩ (ball c R \ s))

Next tactic:
obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := by
    refine' nonempty_diff.2 fun hsub => _
    have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
    rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this
    exact this.not_lt Cardinal.aleph0_lt_continuum
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := 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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
⊢ Set.Nonempty (Ioo l u \ g ⁻¹' s)

Next tactic:
refine' nonempty_diff.2 fun hsub => _
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := by
    refine' nonempty_diff.2 fun hsub => _
    

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
hsub : Ioo l u ⊆ g ⁻¹' s
⊢ False

Next tactic:
have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := by
    refine' nonempty_diff.2 fun hsub => _
    have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
    

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this✝ : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
hsub : Ioo l u ⊆ g ⁻¹' s
this : Set.Countable (Ioo l u)
⊢ False

Next tactic:
rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := by
    refine' nonempty_diff.2 fun hsub => _
    have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
    rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this
    

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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this✝ : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
hsub : Ioo l u ⊆ g ⁻¹' s
this : Cardinal.continuum ≤ Cardinal.aleph0
⊢ False

Next tactic:
exact this.not_lt Cardinal.aleph0_lt_continuum
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$.
-/
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 := by
  have hR : 0 < R := dist_nonneg.trans_lt hw
  suffices w ∈ closure (ball c R \ s) by
    lift R to ℝ≥0 using hR.le
    have A : ContinuousAt (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z) w := by
      have := hasFPowerSeriesOn_cauchy_integral
        ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR
      refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)
      rwa [Metric.emetric_ball_nnreal]
    have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)
    refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)
    intro z hz
    rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd,
      inv_smul_smul₀]
    simp [Real.pi_ne_zero, I_ne_zero]
  refine' mem_closure_iff_nhds.2 fun t ht => _
  -- TODO: generalize to any vector space over `ℝ`
  set g : ℝ → ℂ := fun x => w + ofReal x
  have : Tendsto g (𝓝 0) (𝓝 w) :=
    (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)
  rcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with
    ⟨l, u, hlu₀, hlu_sub⟩
  obtain ⟨x, hx⟩ : (Ioo l u \ g ⁻¹' s).Nonempty := by
    refine' nonempty_diff.2 fun hsub => _
    have : (Ioo l u).Countable :=
      (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub
    rw [← Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu₀.1.trans hlu₀.2)] at this
    exact this.not_lt Cardinal.aleph0_lt_continuum
  

Proof state:
case intro.intro.intro.intro
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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
hR : 0 < R
t : Set ℂ
ht : t ∈ 𝓝 w
g : ℝ → ℂ := fun x => w + ofReal x
this : Tendsto g (𝓝 0) (𝓝 w)
l u : ℝ
hlu₀ : 0 ∈ Ioo l u
hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)
x : ℝ
hx : x ∈ Ioo l u \ g ⁻¹' s
⊢ Set.Nonempty (t ∩ (ball c R \ s))

Next tactic:
exact ⟨g x, (hlu_sub hx.1).1, (hlu_sub hx.1).2, hx.2⟩
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$.
-/
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 := 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
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:
rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
    hs hw hc hd, smul_inv_smul₀]
/-- **Cauchy integral formula**: if `f : ℂ → E` is continuous on a closed disc of radius `R` and is
complex differentiable at all but countably many points of its interior, then for any `w` in this
interior we have $\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=2πif(w)$.
-/
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 := by
  rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
    hs hw hc hd, smul_inv_smul₀]
  

Proof state:
case hc
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
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
⊢ 2 * ↑π * I ≠ 0

Next tactic:
simp [Real.pi_ne_zero, I_ne_zero]
/-- **Cauchy integral formula**: if `f : ℂ → E` is complex differentiable on an open disc and is
continuous on its closure, then for any `w` in this open ball we have
$\frac{1}{2πi}\oint_{|z-c|=R}(z-w)^{-1}f(z)\,dz=f(w)$. -/
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 := 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
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = f w

Next tactic:
have hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (Set.nonempty_of_mem hw).ne_empty)