l3lab/ntp-mathlib huggingface dataset:
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
⊢ IsCountablyGenerated atTop

Next tactic:
by_cases h : ∃ (x : α), IsTop x
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · 

Proof state:
case pos
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ∃ x, IsTop x
⊢ IsCountablyGenerated atTop

Next tactic:
rcases h with ⟨x, hx⟩
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    

Proof state:
case pos.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
x : α
hx : IsTop x
⊢ IsCountablyGenerated atTop

Next tactic:
rw [atTop_eq_pure_of_isTop hx]
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    

Proof state:
case pos.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
x : α
hx : IsTop x
⊢ IsCountablyGenerated (pure x)

Next tactic:
exact isCountablyGenerated_pure x
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · 

Proof state:
case neg
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
⊢ IsCountablyGenerated atTop

Next tactic:
rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    

Proof state:
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
⊢ IsCountablyGenerated atTop

Next tactic:
have : Countable b := by exact Iff.mpr countable_coe_iff b_count
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by 

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
⊢ Countable ↑b

Next tactic:
exact Iff.mpr countable_coe_iff b_count
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    

Proof state:
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
⊢ IsCountablyGenerated atTop

Next tactic:
have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
⊢ ∀ (s : ↑b), ∃ x, x ∈ ↑s

Next tactic:
intro s
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
⊢ ∃ x, x ∈ ↑s

Next tactic:
have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
⊢ ↑s ≠ ∅

Next tactic:
intro H
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ False

Next tactic:
apply b_ne
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ ∅ ∈ b

Next tactic:
convert s.2
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        

Proof state:
case h.e'_4
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
s : ↑b
H : ↑s = ∅
⊢ ∅ = ↑s

Next tactic:
exact H.symm
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Countable ↑b
s : ↑b
this : ↑s ≠ ∅
⊢ ∃ x, x ∈ ↑s

Next tactic:
exact Iff.mp nmem_singleton_empty this
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    

Proof state:
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
A : ∀ (s : ↑b), ∃ x, x ∈ ↑s
⊢ IsCountablyGenerated atTop

Next tactic:
choose a ha using A
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    

Proof state:
case neg.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
⊢ IsCountablyGenerated atTop

Next tactic:
have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      rcases h x with ⟨y, hy⟩
      obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
        hb.exists_subset_of_mem_open hy isOpen_Ioi
      have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)
      have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)
      exact lt_irrefl _ (I.trans_lt J)
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      

Proof state:
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
⊢ atTop = generate (Ici '' range a)

Next tactic:
apply atTop_eq_generate_of_not_bddAbove
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      

Proof state:
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
⊢ ¬BddAbove (range a)

Next tactic:
intro ⟨x, hx⟩
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      

Proof state:
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
h : ¬∃ x, IsTop x
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
⊢ False

Next tactic:
simp only [IsTop, not_exists, not_forall, not_le] at h
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      

Proof state:
case hs
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
h : ∀ (x : α), ∃ x_1, x < x_1
⊢ False

Next tactic:
rcases h x with ⟨y, hy⟩
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      rcases h x with ⟨y, hy⟩
      

Proof state:
case hs.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
h : ∀ (x : α), ∃ x_1, x < x_1
y : α
hy : x < y
⊢ False

Next tactic:
obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
        hb.exists_subset_of_mem_open hy isOpen_Ioi
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      rcases h x with ⟨y, hy⟩
      obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
        hb.exists_subset_of_mem_open hy isOpen_Ioi
      

Proof state:
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
h : ∀ (x : α), ∃ x_1, x < x_1
y : α
hy : x < y
s : Set α
sb : s ∈ b
hs : s ⊆ Ioi x
⊢ False

Next tactic:
have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      rcases h x with ⟨y, hy⟩
      obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
        hb.exists_subset_of_mem_open hy isOpen_Ioi
      have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)
      

Proof state:
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
h : ∀ (x : α), ∃ x_1, x < x_1
y : α
hy : x < y
s : Set α
sb : s ∈ b
hs : s ⊆ Ioi x
I : a { val := s, property := sb } ≤ x
⊢ False

Next tactic:
have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)
instance instIsCountablyGenerated_atTop [SecondCountableTopology α] :
    IsCountablyGenerated (atTop : Filter α) := by
  by_cases h : ∃ (x : α), IsTop x
  · rcases h with ⟨x, hx⟩
    rw [atTop_eq_pure_of_isTop hx]
    exact isCountablyGenerated_pure x
  · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩
    have : Countable b := by exact Iff.mpr countable_coe_iff b_count
    have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by
      intro s
      have : (s : Set α) ≠ ∅ := by
        intro H
        apply b_ne
        convert s.2
        exact H.symm
      exact Iff.mp nmem_singleton_empty this
    choose a ha using A
    have : (atTop : Filter α) = (generate (Ici '' (range a))) := by
      apply atTop_eq_generate_of_not_bddAbove
      intro ⟨x, hx⟩
      simp only [IsTop, not_exists, not_forall, not_le] at h
      rcases h x with ⟨y, hy⟩
      obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x :=
        hb.exists_subset_of_mem_open hy isOpen_Ioi
      have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _)
      have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩)
      

Proof state:
case hs.intro.intro.intro.intro
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
b : Set (Set α)
b_count : Set.Countable b
b_ne : ∅ ∉ b
hb : IsTopologicalBasis b
this : Countable ↑b
a : ↑b → α
ha : ∀ (s : ↑b), a s ∈ ↑s
x : α
hx : x ∈ upperBounds (range a)
h : ∀ (x : α), ∃ x_1, x < x_1
y : α
hy : x < y
s : Set α
sb : s ∈ b
hs : s ⊆ Ioi x
I : a { val := s, property := sb } ≤ x
J : x < a { val := s, property := sb }
⊢ False

Next tactic:
exact lt_irrefl _ (I.trans_lt J)