l3lab/ntp-mathlib huggingface dataset:
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
⊢ v ∈ f '' closedBall z₀ r

Next tactic:
have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by 

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
⊢ ‖f z₀ - v‖ < ε / 2

Next tactic:
simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
⊢ v ∈ f '' closedBall z₀ r

Next tactic:
obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
⊢ ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
⊢ v ∈ f '' closedBall z₀ r

Next tactic:
exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  

Proof state:
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
⊢ v ∈ f '' closedBall z₀ r

Next tactic:
refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
  

Proof state:
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
⊢ f z - v = 0

Next tactic:
have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
  have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
  

Proof state:
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z => f z - v) y
⊢ f z - v = 0

Next tactic:
refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
  have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
  refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_
  

Proof state:
case intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z => f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
⊢ False

Next tactic:
have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by filter_upwards [key] with h; field_simp
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
  have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
  refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_
  have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by 

Proof state:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z => f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
⊢ ∀ᶠ (w : ℂ) in 𝓝 z, f w = f z

Next tactic:
filter_upwards [key] with h
/-- If the modulus of a holomorphic function `f` is bounded below by `ε` on a circle, then its range
contains a disk of radius `ε / 2`. -/
theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
    (hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
    ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by
  /- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
    the function `λ z, ‖f z - v‖`: it is bounded below on the circle, and takes a small value
    at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
    hence that `v` is in the range of `f`. -/
  rintro v hv
  have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
  have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
    continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
  have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
    linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
      norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
  have h5 : ‖f z₀ - v‖ < ε / 2 := by simpa [← dist_eq_norm, dist_comm] using mem_ball.mp hv
  obtain ⟨z, hz1, hz2⟩ : ∃ z ∈ ball z₀ r, IsLocalMin (fun z => ‖f z - v‖) z
  exact exists_isLocalMin_mem_ball h2 (mem_closedBall_self hr.le) fun z hz => h5.trans_le (h4 z hz)
  refine ⟨z, ball_subset_closedBall hz1, sub_eq_zero.mp ?_⟩
  have h6 := h1.differentiableOn.eventually_differentiableAt (isOpen_ball.mem_nhds hz1)
  refine (eventually_eq_or_eq_zero_of_isLocalMin_norm h6 hz2).resolve_left fun key => ?_
  have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by filter_upwards [key] with h; 

Proof state:
case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h✝ : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z => f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
h : ℂ
⊢ f h - v = f z - v → f h = f z

Next tactic:
field_simp