l3lab/ntp-mathlib huggingface dataset:
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    

Proof state:
case refine'_1
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s
⊢ s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)

Next tactic:
convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    

Proof state:
case h.e'_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s
⊢ s = Seminorm.ball (gaugeSeminormFamily 𝕜 E { val := s, property := hs }) 0 1

Next tactic:
rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  

Proof state:
case refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)
⊢ ∃ i, (0 ∈ i ∧ IsOpen i ∧ Balanced 𝕜 i ∧ Convex ℝ i) ∧ id i ⊆ id s

Next tactic:
refine' ⟨s, ⟨_, rfl.subset⟩⟩
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  

Proof state:
case refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)
⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s

Next tactic:
rw [SeminormFamily.basisSets_iff] at hs
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  

Proof state:
case refine'_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
s : Set E
hs : ∃ i r, 0 < r ∧ s = Seminorm.ball (Finset.sup i (gaugeSeminormFamily 𝕜 E)) 0 r
⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s

Next tactic:
rcases hs with ⟨t, r, hr, rfl⟩
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
⊢ 0 ∈ Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r ∧
    IsOpen (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧
      Balanced 𝕜 (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧
        Convex ℝ (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r)

Next tactic:
rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
⊢ 0 ∈ ⋂ i ∈ t, Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r ∧
    IsOpen (⋂ i ∈ t, Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧
      Balanced 𝕜 (⋂ i ∈ t, Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧
        Convex ℝ (⋂ i ∈ t, Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r)

Next tactic:
refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
x✝¹ : AbsConvexOpenSets 𝕜 E
x✝ : x✝¹ ∈ t
⊢ 0 ∈ Seminorm.ball (gaugeSeminormFamily 𝕜 E x✝¹) 0 r

Next tactic:
simp [Seminorm.mem_ball_zero, hr]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)

Next tactic:
have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  have hr' : r = ‖(r : 𝕜)‖ * 1 := by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
⊢ r = ‖↑r‖ * 1

Next tactic:
simp [abs_of_pos hr]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
hr' : r = ‖↑r‖ * 1
⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)

Next tactic:
have hr'' : (r : 𝕜) ≠ 0 := by simp [hr.ne']
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]
  have hr'' : (r : 𝕜) ≠ 0 := by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
hr' : r = ‖↑r‖ * 1
⊢ ↑r ≠ 0

Next tactic:
simp [hr.ne']
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]
  have hr'' : (r : 𝕜) ≠ 0 := by simp [hr.ne']
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
hr' : r = ‖↑r‖ * 1
hr'' : ↑r ≠ 0
⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)

Next tactic:
rw [hr', ← Seminorm.smul_ball_zero hr'', gaugeSeminormFamily_ball]
/-- The topology of a locally convex space is induced by the gauge seminorm family. -/
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) := by
  refine' SeminormFamily.withSeminorms_of_hasBasis _ _
  refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _
  · refine' ⟨s, ⟨_, rfl.subset⟩⟩
    convert (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos
    rw [gaugeSeminormFamily_ball, Subtype.coe_mk]
  refine' ⟨s, ⟨_, rfl.subset⟩⟩
  rw [SeminormFamily.basisSets_iff] at hs
  rcases hs with ⟨t, r, hr, rfl⟩
  rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]
  -- We have to show that the intersection contains zero, is open, balanced, and convex
  refine'
    ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],
      isOpen_biInter_finset fun S _ => _,
      balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,
      convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩
  -- The only nontrivial part is to show that the ball is open
  have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]
  have hr'' : (r : 𝕜) ≠ 0 := by simp [hr.ne']
  rw [hr', ← Seminorm.smul_ball_zero hr'', gaugeSeminormFamily_ball]
  

Proof state:
case refine'_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
inst✝¹⁰ : IsROrC 𝕜
inst✝⁹ : AddCommGroup E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module ℝ E
inst✝⁵ : IsScalarTower ℝ 𝕜 E
inst✝⁴ : ContinuousSMul ℝ E
inst✝³ : TopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : SMulCommClass ℝ 𝕜 E
inst✝ : LocallyConvexSpace ℝ E
t : Finset (AbsConvexOpenSets 𝕜 E)
r : ℝ
hr : 0 < r
S : AbsConvexOpenSets 𝕜 E
x✝ : S ∈ t
hr' : r = ‖↑r‖ * 1
hr'' : ↑r ≠ 0
⊢ IsOpen (↑r • ↑S)

Next tactic:
exact S.coe_isOpen.smul₀ hr''
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
  -- jumps down on the right of `t`. This jump set is countable for any function.
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
⊢ Set.Countable {t | ↑↑μ {a | t ≤ g a} ≠ ↑↑μ {a | t < g a}}

Next tactic:
let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
  -- jumps down on the right of `t`. This jump set is countable for any function.
  let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
F : R → ℝ≥0∞ := fun t => ↑↑μ {a | t ≤ g a}
⊢ Set.Countable {t | ↑↑μ {a | t ≤ g a} ≠ ↑↑μ {a | t < g a}}

Next tactic:
apply (countable_image_gt_image_Ioi F).mono
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
  -- jumps down on the right of `t`. This jump set is countable for any function.
  let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
  apply (countable_image_gt_image_Ioi F).mono
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
F : R → ℝ≥0∞ := fun t => ↑↑μ {a | t ≤ g a}
⊢ {t | ↑↑μ {a | t ≤ g a} ≠ ↑↑μ {a | t < g a}} ⊆ {x | ∃ z < F x, ∀ (y : R), x < y → F y ≤ z}

Next tactic:
intro t ht
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
  -- jumps down on the right of `t`. This jump set is countable for any function.
  let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
  apply (countable_image_gt_image_Ioi F).mono
  intro t ht
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
F : R → ℝ≥0∞ := fun t => ↑↑μ {a | t ≤ g a}
t : R
ht : t ∈ {t | ↑↑μ {a | t ≤ g a} ≠ ↑↑μ {a | t < g a}}
⊢ t ∈ {x | ∃ z < F x, ∀ (y : R), x < y → F y ≤ z}

Next tactic:
have : μ {a | t < g a} < μ {a | t ≤ g a} :=
    lt_of_le_of_ne (measure_mono (fun a ha ↦ le_of_lt ha)) (Ne.symm ht)
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by
  -- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
  -- jumps down on the right of `t`. This jump set is countable for any function.
  let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
  apply (countable_image_gt_image_Ioi F).mono
  intro t ht
  have : μ {a | t < g a} < μ {a | t ≤ g a} :=
    lt_of_le_of_ne (measure_mono (fun a ha ↦ le_of_lt ha)) (Ne.symm ht)
  

Proof state:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
F : R → ℝ≥0∞ := fun t => ↑↑μ {a | t ≤ g a}
t : R
ht : t ∈ {t | ↑↑μ {a | t ≤ g a} ≠ ↑↑μ {a | t < g a}}
this : ↑↑μ {a | t < g a} < ↑↑μ {a | t ≤ g a}
⊢ t ∈ {x | ∃ z < F x, ∀ (y : R), x < y → F y ≤ z}

Next tactic:
exact ⟨μ {a | t < g a}, this, fun s hs ↦ measure_mono (fun a ha ↦ hs.trans_le ha)⟩