l3lab/ntp-mathlib huggingface dataset:
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    

Proof state:
case intro.intro.intro.intro.intro.refine'_1.refine'_1
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
⊢ 0 ≤ (ε / 2)⁻¹
case intro.intro.intro.intro.intro.refine'_1.refine'_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
⊢ 0 ≤ ↑n

Next tactic:
exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by 

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
⊢ 0 ≤ 2

Next tactic:
norm_num
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · 

Proof state:
case intro.intro.intro.intro.intro.refine'_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
by_cases hy : y = 0
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · 

Proof state:
case pos
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y = 0
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
use 0
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      

Proof state:
case h
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y = 0
⊢ dist (f 0) y ≤ 1 / 2 * ‖y‖ ∧ ‖0‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
simp [hy]
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · 

Proof state:
case neg
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      

Proof state:
case neg.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
let δ := ‖d‖ * ‖y‖ / 4
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      let δ := ‖d‖ * ‖y‖ / 4
      

Proof state:
case neg.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      let δ := ‖d‖ * ‖y‖ / 4
      have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by 

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
⊢ 0 < 4

Next tactic:
norm_num
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      let δ := ‖d‖ * ‖y‖ / 4
      have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)
      

Proof state:
case neg.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Next tactic:
have : a + d • y ∈ ball a ε := by
        simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]
/-- First step of the proof of the Banach open mapping theorem (using completeness of `F`):
by Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.
Rescaling everything, it follows that any `y ∈ F` is arbitrarily well approached by
images of elements of norm at most `C * ‖y‖`.
For further use, we will only need such an element whose image
is within distance `‖y‖/2` of `y`, to apply an iterative process. -/
theorem exists_approx_preimage_norm_le (surj : Surjective f) :
    ∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ := by
  have A : ⋃ n : ℕ, closure (f '' ball 0 n) = Set.univ := by
    refine' Subset.antisymm (subset_univ _) fun y _ => _
    rcases surj y with ⟨x, hx⟩
    rcases exists_nat_gt ‖x‖ with ⟨n, hn⟩
    refine' mem_iUnion.2 ⟨n, subset_closure _⟩
    refine' (mem_image _ _ _).2 ⟨x, ⟨_, hx⟩⟩
    rwa [mem_ball, dist_eq_norm, sub_zero]
  have : ∃ (n : ℕ) (x : _), x ∈ interior (closure (f '' ball 0 n)) :=
    nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A
  simp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this
  rcases this with ⟨n, a, ε, ⟨εpos, H⟩⟩
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨(ε / 2)⁻¹ * ‖c‖ * 2 * n, _, fun y => _⟩
  · refine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _
    exacts [inv_nonneg.2 (div_nonneg (le_of_lt εpos) (by norm_num)), n.cast_nonneg]
  · by_cases hy : y = 0
    · use 0
      simp [hy]
    · rcases rescale_to_shell hc (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
      let δ := ‖d‖ * ‖y‖ / 4
      have δpos : 0 < δ := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)
      have : a + d • y ∈ ball a ε := by
        

Proof state:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E →L[𝕜] F
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : ¬y = 0
d : 𝕜
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
⊢ a + d • y ∈ ball a ε

Next tactic:
simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]