l3lab/ntp-mathlib huggingface dataset:
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Next tactic:
have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Next tactic:
have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
⊢ ∃ x, x ∉ F

Next tactic:
contrapose! h
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
⊢ FiniteDimensional 𝕜 E

Next tactic:
have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
⊢ ⊤ = F

Next tactic:
ext x
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      

Proof state:
case h
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
x : E
⊢ x ∈ ⊤ ↔ x ∈ F

Next tactic:
simp [h]
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
this : ⊤ = F
⊢ FiniteDimensional 𝕜 E

Next tactic:
have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by 

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
this : ⊤ = F
⊢ FiniteDimensional 𝕜 ↥⊤

Next tactic:
rwa [this]
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝¹ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
h : ∀ (x : E), x ∈ F
this✝ : ⊤ = F
this : FiniteDimensional 𝕜 ↥⊤
⊢ FiniteDimensional 𝕜 E

Next tactic:
refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Next tactic:
obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
  

Proof state:
case intro.intro
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Next tactic:
have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    intro y hy
    rw [← norm_neg]
    simpa using hx y hy
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
  have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
⊢ ∀ y ∈ F, 1 ≤ ‖y - x‖

Next tactic:
intro y hy
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
  have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    intro y hy
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
y : E
hy : y ∈ F
⊢ 1 ≤ ‖y - x‖

Next tactic:
rw [← norm_neg]
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
  have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    intro y hy
    rw [← norm_neg]
    

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
y : E
hy : y ∈ F
⊢ 1 ≤ ‖-(y - x)‖

Next tactic:
simpa using hx y hy
/-- In an infinite dimensional space, given a finite number of points, one may find a point
with norm at most `R` which is at distance at least `1` of all these points. -/
theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by
  let F := Submodule.span 𝕜 (s : Set E)
  haveI : FiniteDimensional 𝕜 F :=
    Module.finite_def.2
      ((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
  have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
  have : ∃ x, x ∉ F := by
    contrapose! h
    have : (⊤ : Submodule 𝕜 E) = F := by
      ext x
      simp [h]
    have : FiniteDimensional 𝕜 (⊤ : Submodule 𝕜 E) := by rwa [this]
    refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))
  obtain ⟨x, xR, hx⟩ : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y : E, y ∈ F → 1 ≤ ‖x - y‖ :=
    riesz_lemma_of_norm_lt hc hR Fclosed this
  have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    intro y hy
    rw [← norm_neg]
    simpa using hx y hy
  

Proof state:
case intro.intro
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F✝ : Type w
inst✝⁷ : NormedAddCommGroup F✝
inst✝⁶ : NormedSpace 𝕜 F✝
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
hx' : ∀ y ∈ F, 1 ≤ ‖y - x‖
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Next tactic:
exact ⟨x, xR, fun y hy => hx' _ (Submodule.subset_span hy)⟩
/-- In an infinite-dimensional normed space, there exists a sequence of points which are all
bounded by `R` and at distance at least `1`. For a version not assuming `c` and `R`, see
`exists_seq_norm_le_one_le_norm_sub`. -/
theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
    (h : ¬FiniteDimensional 𝕜 E) :
    ∃ f : ℕ → E, (∀ n, ‖f n‖ ≤ R) ∧ ∀ m n, m ≠ n → 1 ≤ ‖f m - f n‖ := by
  

Proof state:
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type w
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
F' : Type x
inst✝⁵ : AddCommGroup F'
inst✝⁴ : Module 𝕜 F'
inst✝³ : TopologicalSpace F'
inst✝² : TopologicalAddGroup F'
inst✝¹ : ContinuousSMul 𝕜 F'
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
⊢ ∃ f, (∀ (n : ℕ), ‖f n‖ ≤ R) ∧ ∀ (m n : ℕ), m ≠ n → 1 ≤ ‖f m - f n‖

Next tactic:
have : IsSymm E fun x y : E => 1 ≤ ‖x - y‖ := by
    constructor
    intro x y hxy
    rw [← norm_neg]
    simpa