l3lab/ntp-mathlib huggingface dataset:
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  

Proof state:
case intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ
ε0 : 0 < ε
hε : ∀ (x : E), ‖f x‖ < ε → ‖x‖ < 1
⊢ ∃ K, AntilipschitzWith K ⇑f

Next tactic:
lift ε to ℝ≥0 using ε0.le
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
⊢ ∃ K, AntilipschitzWith K ⇑f

Next tactic:
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  

Proof state:
case intro.intro.intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
⊢ ∃ K, AntilipschitzWith K ⇑f

Next tactic:
refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  

Proof state:
case intro.intro.intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
by_cases hx : f x = 0
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · 

Proof state:
case pos
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : f x = 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
rw [← hx] at hf
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    

Proof state:
case pos
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hf : comap (⇑f) (𝓝 (f x)) ≤ 𝓝 0
hx : f x = 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    

Proof state:
case pos
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
hf : comap (⇑f) (𝓝 (f 0)) ≤ 𝓝 0
hx : f 0 = 0
⊢ ‖0‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f 0‖

Next tactic:
exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  

Proof state:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : ¬f x = 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  

Proof state:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
rw [← h.1] at hc
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  rw [← h.1] at hc
  

Proof state:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  rw [← h.1] at hc
  rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩
  

Proof state:
case neg.intro.intro.intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε
hle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  rw [← h.1] at hc
  rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩
  simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle
  

Proof state:
case neg.intro.intro.intro.intro
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖f (c ^ n • x)‖ < ↑ε
hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖
⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖

Next tactic:
calc
    ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by
      rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]
    _ ≤ ‖c ^ n‖⁻¹ * 1 := (mul_le_mul_of_nonneg_left (hε _ hlt).le (inv_nonneg.2 (norm_nonneg _)))
    _ ≤ ε⁻¹ * ‖c‖ * ‖f x‖ := by rwa [mul_one]
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  rw [← h.1] at hc
  rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩
  simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle
  calc
    ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by
      

Proof state:
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖f (c ^ n • x)‖ < ↑ε
hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖
⊢ ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖

Next tactic:
rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]
theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F)
    (hf : (𝓝 0).comap f ≤ 𝓝 0) : ∃ K, AntilipschitzWith K f := by
  rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with ⟨ε, ε0, hε⟩
  simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hε
  lift ε to ℝ≥0 using ε0.le
  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
  refine' ⟨ε⁻¹ * ‖c‖₊, AddMonoidHomClass.antilipschitz_of_bound f fun x => _⟩
  by_cases hx : f x = 0
  · rw [← hx] at hf
    obtain rfl : x = 0 := Specializes.eq (specializes_iff_pure.2 <|
      ((Filter.tendsto_pure_pure _ _).mono_right (pure_le_nhds _)).le_comap.trans hf)
    exact norm_zero.trans_le (mul_nonneg (NNReal.coe_nonneg _) (norm_nonneg _))
  have hc₀ : c ≠ 0 := norm_pos_iff.1 (one_pos.trans hc)
  rw [← h.1] at hc
  rcases rescale_to_shell_zpow hc ε0 hx with ⟨n, -, hlt, -, hle⟩
  simp only [← map_zpow₀, h.1, ← map_smulₛₗ] at hlt hle
  calc
    ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖ := by
      rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]
    _ ≤ ‖c ^ n‖⁻¹ * 1 := (mul_le_mul_of_nonneg_left (hε _ hlt).le (inv_nonneg.2 (norm_nonneg _)))
    _ ≤ ε⁻¹ * ‖c‖ * ‖f x‖ := by 

Proof state:
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedAddCommGroup Fₗ
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : NontriviallyNormedField 𝕜₃
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : NormedSpace 𝕜₃ G
inst✝ : NormedSpace 𝕜 Fₗ
c✝ : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ g : E →SL[σ₁₂] F
x✝ y z : E
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖f (c ^ n • x)‖ < ↑ε
hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖
⊢ ‖c ^ n‖⁻¹ * 1 ≤ ↑ε⁻¹ * ‖c‖ * ‖f x‖

Next tactic:
rwa [mul_one]
/-- An operator is zero iff its norm vanishes. -/
theorem op_norm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 :=
  Iff.intro
    (fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
      (calc
        _ ≤ ‖f‖ * ‖x‖ := le_op_norm _ _
        _ = _ := by 

Proof state:
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
inst✝¹ : NormedSpace 𝕜 Fₗ
c : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f g : E →SL[σ₁₂] F
x✝ y z : E
inst✝ : RingHomIsometric σ₁₂
hn : ‖f‖ = 0
x : E
⊢ ‖f‖ * ‖x‖ = 0

Next tactic:
rw [hn, zero_mul]
/-- An operator is zero iff its norm vanishes. -/
theorem op_norm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 :=
  Iff.intro
    (fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
      (calc
        _ ≤ ‖f‖ * ‖x‖ := le_op_norm _ _
        _ = _ := by rw [hn, zero_mul]))
    (by
      

Proof state:
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : NormedAddCommGroup Fₗ
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
inst✝¹ : NormedSpace 𝕜 Fₗ
c : 𝕜
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f g : E →SL[σ₁₂] F
x y z : E
inst✝ : RingHomIsometric σ₁₂
⊢ f = 0 → ‖f‖ = 0

Next tactic:
rintro rfl