l3lab/ntp-mathlib huggingface dataset:
/-- Construct a realizer for the sup of two filters -/
protected def sup {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊔ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∪ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ union_subset_union (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ union_subset_union (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    filter_eq <| Set.ext fun _ ↦ by 

Proof state:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
F : Realizer f
G : Realizer g
x✝ : Set α
⊢ x✝ ∈
      (CFilter.toFilter
          {
            f := fun x =>
              match x with
              | (s, t) => CFilter.f F.F s ∪ CFilter.f G.F t,
            pt := (F.F.pt, G.F.pt),
            inf := fun x x_1 =>
              match x with
              | (a, a') =>
                match x_1 with
                | (b, b') => (CFilter.inf F.F a b, CFilter.inf G.F a' b'),
            inf_le_left :=
              (_ :
                ∀ (x x_1 : F.σ × G.σ),
                  CFilter.f F.F (CFilter.inf F.F x.1 x_1.1) ∪ CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                    CFilter.f F.F x.1 ∪ CFilter.f G.F x.2),
            inf_le_right :=
              (_ :
                ∀ (x x_1 : F.σ × G.σ),
                  CFilter.f F.F (CFilter.inf F.F x.1 x_1.1) ∪ CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                    CFilter.f F.F x_1.1 ∪ CFilter.f G.F x_1.2) }).sets ↔
    x✝ ∈ (f ⊔ g).sets

Next tactic:
cases F
/-- Construct a realizer for the sup of two filters -/
protected def sup {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊔ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∪ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ union_subset_union (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ union_subset_union (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    filter_eq <| Set.ext fun _ ↦ by cases F; 

Proof state:
case mk
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
G : Realizer g
x✝ : Set α
σ✝ : Type ?u.21233
F✝ : CFilter (Set α) σ✝
eq✝ : CFilter.toFilter F✝ = f
⊢ x✝ ∈
      (CFilter.toFilter
          {
            f := fun x =>
              match x with
              | (s, t) => CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F s ∪ CFilter.f G.F t,
            pt := ({ σ := σ✝, F := F✝, eq := eq✝ }.F.pt, G.F.pt),
            inf := fun x x_1 =>
              match x with
              | (a, a') =>
                match x_1 with
                | (b, b') => (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F a b, CFilter.inf G.F a' b'),
            inf_le_left :=
              (_ :
                ∀ (x x_1 : { σ := σ✝, F := F✝, eq := eq✝ }.σ × G.σ),
                  CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                        (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 x_1.1) ∪
                      CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                    CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 ∪ CFilter.f G.F x.2),
            inf_le_right :=
              (_ :
                ∀ (x x_1 : { σ := σ✝, F := F✝, eq := eq✝ }.σ × G.σ),
                  CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                        (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 x_1.1) ∪
                      CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                    CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x_1.1 ∪ CFilter.f G.F x_1.2) }).sets ↔
    x✝ ∈ (f ⊔ g).sets

Next tactic:
cases G
/-- Construct a realizer for the sup of two filters -/
protected def sup {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊔ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∪ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ union_subset_union (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ union_subset_union (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    filter_eq <| Set.ext fun _ ↦ by cases F; cases G; 

Proof state:
case mk.mk
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
x✝ : Set α
σ✝¹ : Type ?u.21233
F✝¹ : CFilter (Set α) σ✝¹
eq✝¹ : CFilter.toFilter F✝¹ = f
σ✝ : Type ?u.21288
F✝ : CFilter (Set α) σ✝
eq✝ : CFilter.toFilter F✝ = g
⊢ x✝ ∈
      (CFilter.toFilter
          {
            f := fun x =>
              match x with
              | (s, t) =>
                CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F s ∪ CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F t,
            pt := ({ σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F.pt, { σ := σ✝, F := F✝, eq := eq✝ }.F.pt),
            inf := fun x x_1 =>
              match x with
              | (a, a') =>
                match x_1 with
                | (b, b') =>
                  (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F a b,
                    CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F a' b'),
            inf_le_left :=
              (_ :
                ∀ (x x_1 : { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.σ × { σ := σ✝, F := F✝, eq := eq✝ }.σ),
                  CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F
                        (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 x_1.1) ∪
                      CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                        (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.2 x_1.2) ⊆
                    CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 ∪
                      CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x.2),
            inf_le_right :=
              (_ :
                ∀ (x x_1 : { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.σ × { σ := σ✝, F := F✝, eq := eq✝ }.σ),
                  CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F
                        (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 x_1.1) ∪
                      CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                        (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.2 x_1.2) ⊆
                    CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x_1.1 ∪
                      CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x_1.2) }).sets ↔
    x✝ ∈ (f ⊔ g).sets

Next tactic:
substs f g
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      

Proof state:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
F : Realizer f
G : Realizer g
⊢ CFilter.toFilter
      {
        f := fun x =>
          match x with
          | (s, t) => CFilter.f F.F s ∩ CFilter.f G.F t,
        pt := (F.F.pt, G.F.pt),
        inf := fun x x_1 =>
          match x with
          | (a, a') =>
            match x_1 with
            | (b, b') => (CFilter.inf F.F a b, CFilter.inf G.F a' b'),
        inf_le_left :=
          (_ :
            ∀ (x x_1 : F.σ × G.σ),
              CFilter.f F.F (CFilter.inf F.F x.1 x_1.1) ∩ CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                CFilter.f F.F x.1 ∩ CFilter.f G.F x.2),
        inf_le_right :=
          (_ :
            ∀ (x x_1 : F.σ × G.σ),
              CFilter.f F.F (CFilter.inf F.F x.1 x_1.1) ∩ CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                CFilter.f F.F x_1.1 ∩ CFilter.f G.F x_1.2) } =
    f ⊓ g

Next tactic:
cases F
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; 

Proof state:
case mk
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
G : Realizer g
σ✝ : Type ?u.33500
F✝ : CFilter (Set α) σ✝
eq✝ : CFilter.toFilter F✝ = f
⊢ CFilter.toFilter
      {
        f := fun x =>
          match x with
          | (s, t) => CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F s ∩ CFilter.f G.F t,
        pt := ({ σ := σ✝, F := F✝, eq := eq✝ }.F.pt, G.F.pt),
        inf := fun x x_1 =>
          match x with
          | (a, a') =>
            match x_1 with
            | (b, b') => (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F a b, CFilter.inf G.F a' b'),
        inf_le_left :=
          (_ :
            ∀ (x x_1 : { σ := σ✝, F := F✝, eq := eq✝ }.σ × G.σ),
              CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 x_1.1) ∩
                  CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 ∩ CFilter.f G.F x.2),
        inf_le_right :=
          (_ :
            ∀ (x x_1 : { σ := σ✝, F := F✝, eq := eq✝ }.σ × G.σ),
              CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.1 x_1.1) ∩
                  CFilter.f G.F (CFilter.inf G.F x.2 x_1.2) ⊆
                CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x_1.1 ∩ CFilter.f G.F x_1.2) } =
    f ⊓ g

Next tactic:
cases G
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; 

Proof state:
case mk.mk
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
eq✝¹ : CFilter.toFilter F✝¹ = f
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
eq✝ : CFilter.toFilter F✝ = g
⊢ CFilter.toFilter
      {
        f := fun x =>
          match x with
          | (s, t) => CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F s ∩ CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F t,
        pt := ({ σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F.pt, { σ := σ✝, F := F✝, eq := eq✝ }.F.pt),
        inf := fun x x_1 =>
          match x with
          | (a, a') =>
            match x_1 with
            | (b, b') =>
              (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F a b,
                CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F a' b'),
        inf_le_left :=
          (_ :
            ∀ (x x_1 : { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.σ × { σ := σ✝, F := F✝, eq := eq✝ }.σ),
              CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F
                    (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 x_1.1) ∩
                  CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                    (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.2 x_1.2) ⊆
                CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 ∩ CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x.2),
        inf_le_right :=
          (_ :
            ∀ (x x_1 : { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.σ × { σ := σ✝, F := F✝, eq := eq✝ }.σ),
              CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F
                    (CFilter.inf { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x.1 x_1.1) ∩
                  CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F
                    (CFilter.inf { σ := σ✝, F := F✝, eq := eq✝ }.F x.2 x_1.2) ⊆
                CFilter.f { σ := σ✝¹, F := F✝¹, eq := eq✝¹ }.F x_1.1 ∩
                  CFilter.f { σ := σ✝, F := F✝, eq := eq✝ }.F x_1.2) } =
    f ⊓ g

Next tactic:
substs f g
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; substs f g; simp only [CFilter.toFilter, Prod.exists]; 

Proof state:
case mk.mk
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
⊢ { sets := {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a},
      univ_sets := (_ : univ ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
      sets_of_superset :=
        (_ :
          ∀ {x y : Set α},
            x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
              x ⊆ y → y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
      inter_sets :=
        (_ :
          ∀ {x y : Set α},
            x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
              y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                x ∩ y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}) } =
    { sets := {a | ∃ b, CFilter.f F✝¹ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝¹ b ⊆ univ),
        sets_of_superset :=
          (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} →
                y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}) } ⊓
      { sets := {a | ∃ b, CFilter.f F✝ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝ b ⊆ univ),
        sets_of_superset :=
          (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} →
                y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}) }

Next tactic:
ext
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; substs f g; simp only [CFilter.toFilter, Prod.exists]; ext
      

Proof state:
case mk.mk.a
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
s✝ : Set α
⊢ s✝ ∈
      { sets := {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a},
        univ_sets := (_ : univ ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
        sets_of_superset :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                x ⊆ y → y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                  x ∩ y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}) } ↔
    s✝ ∈
      { sets := {a | ∃ b, CFilter.f F✝¹ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝¹ b ⊆ univ),
          sets_of_superset :=
            (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}),
          inter_sets :=
            (_ :
              ∀ {x y : Set α},
                x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} →
                  y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}) } ⊓
        { sets := {a | ∃ b, CFilter.f F✝ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝ b ⊆ univ),
          sets_of_superset :=
            (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}),
          inter_sets :=
            (_ :
              ∀ {x y : Set α},
                x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} →
                  y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}) }

Next tactic:
constructor
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; substs f g; simp only [CFilter.toFilter, Prod.exists]; ext
      constructor
      · 

Proof state:
case mk.mk.a.mp
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
s✝ : Set α
⊢ s✝ ∈
      { sets := {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a},
        univ_sets := (_ : univ ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
        sets_of_superset :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                x ⊆ y → y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a} →
                  x ∩ y ∈ {a | ∃ a_1 b, CFilter.f F✝¹ a_1 ∩ CFilter.f F✝ b ⊆ a}) } →
    s✝ ∈
      { sets := {a | ∃ b, CFilter.f F✝¹ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝¹ b ⊆ univ),
          sets_of_superset :=
            (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}),
          inter_sets :=
            (_ :
              ∀ {x y : Set α},
                x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} →
                  y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}) } ⊓
        { sets := {a | ∃ b, CFilter.f F✝ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝ b ⊆ univ),
          sets_of_superset :=
            (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}),
          inter_sets :=
            (_ :
              ∀ {x y : Set α},
                x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} →
                  y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}) }

Next tactic:
rintro ⟨s, t, h⟩
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; substs f g; simp only [CFilter.toFilter, Prod.exists]; ext
      constructor
      · rintro ⟨s, t, h⟩
        

Proof state:
case mk.mk.a.mp.intro.intro
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
s✝ : Set α
s : σ✝¹
t : σ✝
h : CFilter.f F✝¹ s ∩ CFilter.f F✝ t ⊆ s✝
⊢ s✝ ∈
    { sets := {a | ∃ b, CFilter.f F✝¹ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝¹ b ⊆ univ),
        sets_of_superset :=
          (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} →
                y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}) } ⊓
      { sets := {a | ∃ b, CFilter.f F✝ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝ b ⊆ univ),
        sets_of_superset :=
          (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}),
        inter_sets :=
          (_ :
            ∀ {x y : Set α},
              x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} →
                y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}) }

Next tactic:
apply mem_inf_of_inter _ _ h
/-- Construct a realizer for the inf of two filters -/
protected def inf {f g : Filter α} (F : f.Realizer) (G : g.Realizer) : (f ⊓ g).Realizer :=
  ⟨F.σ × G.σ,
    { f := fun ⟨s, t⟩ ↦ F.F s ∩ G.F t
      pt := (F.F.pt, G.F.pt)
      inf := fun ⟨a, a'⟩ ⟨b, b'⟩ ↦ (F.F.inf a b, G.F.inf a' b')
      inf_le_left := fun _ _ ↦ inter_subset_inter (F.F.inf_le_left _ _) (G.F.inf_le_left _ _)
      inf_le_right := fun _ _ ↦ inter_subset_inter (F.F.inf_le_right _ _) (G.F.inf_le_right _ _) },
    by
      cases F; cases G; substs f g; simp only [CFilter.toFilter, Prod.exists]; ext
      constructor
      · rintro ⟨s, t, h⟩
        apply mem_inf_of_inter _ _ h
        

Proof state:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
s✝ : Set α
s : σ✝¹
t : σ✝
h : CFilter.f F✝¹ s ∩ CFilter.f F✝ t ⊆ s✝
⊢ CFilter.f F✝¹ s ∈
    { sets := {a | ∃ b, CFilter.f F✝¹ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝¹ b ⊆ univ),
      sets_of_superset :=
        (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}),
      inter_sets :=
        (_ :
          ∀ {x y : Set α},
            x ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} →
              y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝¹ b ⊆ a}) }
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ✝¹ : Type ?u.33500
F✝¹ : CFilter (Set α) σ✝¹
σ✝ : Type ?u.33555
F✝ : CFilter (Set α) σ✝
s✝ : Set α
s : σ✝¹
t : σ✝
h : CFilter.f F✝¹ s ∩ CFilter.f F✝ t ⊆ s✝
⊢ CFilter.f F✝ t ∈
    { sets := {a | ∃ b, CFilter.f F✝ b ⊆ a}, univ_sets := (_ : ∃ b, CFilter.f F✝ b ⊆ univ),
      sets_of_superset :=
        (_ : ∀ {x y : Set α}, x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ⊆ y → y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}),
      inter_sets :=
        (_ :
          ∀ {x y : Set α},
            x ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} →
              y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a} → x ∩ y ∈ {a | ∃ b, CFilter.f F✝ b ⊆ a}) }

Next tactic:
use s