awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
σ : Type u_3
τ : Type u_4
inst : PartialOrder α
E : σ ≃ τ
F : CFilter α σ
a : τ
⊢ (ofEquiv E F).f a = F.f ((E.symm : (a : τ) → σ) a)
have mk : ∀ (f : σ → α) (pt : σ) (inf : σ → σ → σ) (inf_le_left : ∀ (a b : σ), f (inf a b) ≤ f a)
  (inf_le_right : ∀ (a b : σ), f (inf a b) ≤ f b),
  (ofEquiv E { f := f, pt := pt, inf := inf, inf_le_left := inf_le_left, inf_le_right := inf_le_right : CFilter α σ }).f
      a =
    { f := f, pt := pt, inf := inf, inf_le_left := inf_le_left, inf_le_right := inf_le_right : CFilter α σ }.f
      ((E.symm : (a : τ) → σ) a)
Tactic: cases F
Cleared hypotheses: F
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
f : Filter α
F : Realizer.{u_1, u_5} f
a : Set α
⊢ a ∈ f ↔ ∃ (b : F.σ), F.F.f b ⊆ a
have mk : ∀ (σ : Type u_5) (F : CFilter.{u_1, u_5} (Set α) σ) (eq : F.toFilter = f),
  a ∈ f ↔
    ∃ (b : { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ),
      { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.f b ⊆ a
Tactic: cases F
Cleared hypotheses: F
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
f : Filter α
a : Set α
σ : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ
eq : F.toFilter = f
⊢ a ∈ f ↔
    ∃ (b : { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ),
      { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.f b ⊆ a
have mk : a ∈ F.toFilter ↔
  ∃ (b : { σ := σ, F := F, eq := sorry : Realizer.{u_1, u_5} F.toFilter }.σ),
    { σ := σ, F := F, eq := sorry : Realizer.{u_1, u_5} F.toFilter }.F.f b ⊆ a
Tactic: subst f
Cleared hypotheses: f, eq
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f : Filter α
F : Realizer.{u_1, u_5} f
E : Equiv.{u_5 + 1, u_4 + 1} F.σ τ
⊢ (CFilter.ofEquiv E F.F).toFilter = f
have refine : (CFilter.ofEquiv E F.F).toFilter = F.F.toFilter
Tactic: refine Eq.trans ?_ F.eq
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
m : α → β
f : Filter α
F : Realizer.{u_1, u_5} f
x : Set β
⊢ Membership.mem (γ := Set (Set β))
      (sets
        (CFilter.toFilter
          { f := fun (s : F.σ) ↦ m '' F.F.f s, pt := F.F.pt, inf := F.F.inf, inf_le_left := ⋯, inf_le_right := ⋯ :
            CFilter.{u_2, u_5} (Set β) F.σ }))
      x ↔
    x ∈ (map m f).sets
have simp : (∃ (b : F.σ), F.F.f b ⊆ m ⁻¹' x) ↔ m ⁻¹' x ∈ f
Tactic: simp only [CFilter.toFilter, image_subset_iff, mem_setOf_eq, Filter.mem_sets, mem_map]
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
m : α → β
f : Filter α
F : Realizer.{u_1, u_5} f
x : Set β
⊢ (∃ (b : F.σ), F.F.f b ⊆ m ⁻¹' x) ↔ m ⁻¹' x ∈ f
have rw : (∃ (b : F.σ), F.F.f b ⊆ m ⁻¹' x) ↔ ∃ (b : F.σ), F.F.f b ⊆ m ⁻¹' x
Tactic: rw [F.mem_sets]
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
m : α → β
f : Filter β
F : Realizer.{u_2, u_5} f
x : Set α
⊢ Membership.mem (γ := Set (Set α))
      (sets
        (CFilter.toFilter
          { f := fun (s : F.σ) ↦ m ⁻¹' F.F.f s, pt := F.F.pt, inf := F.F.inf, inf_le_left := ⋯, inf_le_right := ⋯ :
            CFilter.{u_1, u_5} (Set α) F.σ }))
      x ↔
    x ∈ (comap m f).sets
have mk : ∀ (σ : Type u_5) (F : CFilter.{u_2, u_5} (Set β) σ) (eq : F.toFilter = f),
  Membership.mem (γ := Set (Set α))
      (sets
        (CFilter.toFilter
          {
            f := fun (s : { σ := σ, F := F, eq := eq : Realizer.{u_2, u_5} f }.σ) ↦
              m ⁻¹' { σ := σ, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.f s,
            pt := { σ := σ, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.pt,
            inf := { σ := σ, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.inf, inf_le_left := sorry, inf_le_right := sorry :
            CFilter.{u_1, u_5} (Set α) { σ := σ, F := F, eq := eq : Realizer.{u_2, u_5} f }.σ }))
      x ↔
    x ∈ (comap m f).sets
Tactic: cases F
Cleared hypotheses: F
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
m : α → β
f : Filter β
x : Set α
σ_1 : Type u_5
F : CFilter.{u_2, u_5} (Set β) σ_1
eq : F.toFilter = f
⊢ Membership.mem (γ := Set (Set α))
      (sets
        (CFilter.toFilter
          {
            f := fun (s : { σ := σ_1, F := F, eq := eq : Realizer.{u_2, u_5} f }.σ) ↦
              m ⁻¹' { σ := σ_1, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.f s,
            pt := { σ := σ_1, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.pt,
            inf := { σ := σ_1, F := F, eq := eq : Realizer.{u_2, u_5} f }.F.inf, inf_le_left := ⋯, inf_le_right := ⋯ :
            CFilter.{u_1, u_5} (Set α) { σ := σ_1, F := F, eq := eq : Realizer.{u_2, u_5} f }.σ }))
      x ↔
    x ∈ (comap m f).sets
have mk : Membership.mem (γ := Set (Set α))
    (sets
      (CFilter.toFilter
        {
          f := fun (s : { σ := σ_1, F := F, eq := sorry : Realizer.{u_2, u_5} F.toFilter }.σ) ↦
            m ⁻¹' { σ := σ_1, F := F, eq := sorry : Realizer.{u_2, u_5} F.toFilter }.F.f s,
          pt := { σ := σ_1, F := F, eq := sorry : Realizer.{u_2, u_5} F.toFilter }.F.pt,
          inf := { σ := σ_1, F := F, eq := sorry : Realizer.{u_2, u_5} F.toFilter }.F.inf, inf_le_left := sorry,
          inf_le_right := sorry :
          CFilter.{u_1, u_5} (Set α) { σ := σ_1, F := F, eq := sorry : Realizer.{u_2, u_5} F.toFilter }.σ }))
    x ↔
  x ∈ (comap m F.toFilter).sets
Tactic: subst f
Cleared hypotheses: f, eq
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
F : Realizer.{u_1, u_5} f
G : Realizer.{u_1, u_6} g
x : Set α
⊢ Membership.mem (γ := Set (Set α))
      (sets
        (CFilter.toFilter
          {
            f := fun (x : Prod.{u_5, u_6} F.σ G.σ) ↦
              match (motive := Prod.{u_5, u_6} F.σ G.σ → Set α) x with
              | (s, t) => F.F.f s ∪ G.F.f t,
            pt := (F.F.pt, G.F.pt),
            inf := fun (x x_1 : Prod.{u_5, u_6} F.σ G.σ) ↦
              match (motive := Prod.{u_5, u_6} F.σ G.σ → Prod.{u_5, u_6} F.σ G.σ) x with
              | (a, a') =>
                match (motive := Prod.{u_5, u_6} F.σ G.σ → Prod.{u_5, u_6} F.σ G.σ) x_1 with
                | (b, b') => (F.F.inf a b, G.F.inf a' b'),
            inf_le_left := ⋯, inf_le_right := ⋯ : CFilter (Set α) (Prod.{u_5, u_6} F.σ G.σ) }))
      x ↔
    x ∈ (f ⊔ g).sets
have mk : ∀ (σ : Type u_5) (F : CFilter.{u_1, u_5} (Set α) σ) (eq : F.toFilter = f),
  Membership.mem (γ := Set (Set α))
      (sets
        (CFilter.toFilter
          {
            f := fun (x : Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) ↦
              match (motive := Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ → Set α)
                x with
              | (s, t) => { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.f s ∪ G.F.f t,
            pt := ({ σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.pt, G.F.pt),
            inf := fun (x x_1 : Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) ↦
              match (motive :=
                Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ →
                  Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ)
                x with
              | (a, a') =>
                match (motive :=
                  Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ →
                    Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ)
                  x_1 with
                | (b, b') => ({ σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.inf a b, G.F.inf a' b'),
            inf_le_left := sorry, inf_le_right := sorry :
            CFilter (Set α) (Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) }))
      x ↔
    x ∈ (f ⊔ g).sets
Tactic: cases F
Cleared hypotheses: F
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f g : Filter α
F : Realizer.{u_1, u_5} f
G : Realizer.{u_1, u_6} g
⊢ CFilter.toFilter
      {
        f := fun (x : Prod.{u_5, u_6} F.σ G.σ) ↦
          match (motive := Prod.{u_5, u_6} F.σ G.σ → Set α) x with
          | (s, t) => F.F.f s ∩ G.F.f t,
        pt := (F.F.pt, G.F.pt),
        inf := fun (x x_1 : Prod.{u_5, u_6} F.σ G.σ) ↦
          match (motive := Prod.{u_5, u_6} F.σ G.σ → Prod.{u_5, u_6} F.σ G.σ) x with
          | (a, a') =>
            match (motive := Prod.{u_5, u_6} F.σ G.σ → Prod.{u_5, u_6} F.σ G.σ) x_1 with
            | (b, b') => (F.F.inf a b, G.F.inf a' b'),
        inf_le_left := ⋯, inf_le_right := ⋯ : CFilter (Set α) (Prod.{u_5, u_6} F.σ G.σ) } =
    f ⊓ g
have mk : ∀ (σ : Type u_5) (F : CFilter.{u_1, u_5} (Set α) σ) (eq : F.toFilter = f),
  CFilter.toFilter
      {
        f := fun (x : Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) ↦
          match (motive := Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ → Set α) x with
          | (s, t) => { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.f s ∩ G.F.f t,
        pt := ({ σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.pt, G.F.pt),
        inf := fun (x x_1 : Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) ↦
          match (motive :=
            Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ →
              Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ)
            x with
          | (a, a') =>
            match (motive :=
              Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ →
                Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ)
              x_1 with
            | (b, b') => ({ σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.F.inf a b, G.F.inf a' b'),
        inf_le_left := sorry, inf_le_right := sorry :
        CFilter (Set α) (Prod.{u_5, u_6} { σ := σ, F := F, eq := eq : Realizer.{u_1, u_5} f }.σ G.σ) } =
    f ⊓ g
Tactic: cases F
Cleared hypotheses: F
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
⊢ CFilter.toFilter
      {
        f :=
          fun
            (x :
              Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ) ↦
          match (motive :=
            Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ →
              Set α)
            x with
          | (s, t) =>
            { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.F.f s ∩
              { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.F.f t,
        pt :=
          ({ σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.F.pt,
            { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.F.pt),
        inf :=
          fun
            (x x_1 :
              Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ) ↦
          match (motive :=
            Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ →
              Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ)
            x with
          | (a, a') =>
            match (motive :=
              Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                  { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ →
                Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
                  { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ)
              x_1 with
            | (b, b') =>
              ({ σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.F.inf a b,
                { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.F.inf a' b'),
        inf_le_left := ⋯, inf_le_right := ⋯ :
        CFilter (Set α)
          (Prod.{u_5, u_6} { σ := σ_1, F := F, eq := ⋯ : Realizer.{u_1, u_5} F.toFilter }.σ
            { σ := σ_2, F := F_1, eq := ⋯ : Realizer.{u_1, u_6} F_1.toFilter }.σ) } =
    F.toFilter ⊓ F_1.toFilter
have mk : { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry,
    inter_sets := sorry : Filter α } =
  { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry } ⊓
    { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry }
Tactic: simp only [CFilter.toFilter, Prod.exists]
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
⊢ { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯,
      inter_sets := ⋯ : Filter α } =
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ } ⊓
      { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ }
have mk : ∀ (s : Set α),
  s ∈
      { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry,
        inter_sets := sorry : Filter α } ↔
    s ∈
      { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
          Filter α } ⊓
        { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
          Filter α }
Tactic: ext
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s : Set α
⊢ s ∈
      { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯,
        inter_sets := ⋯ : Filter α } ↔
    s ∈
      { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α } ⊓
        { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α }
have mk : s ∈
    { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry,
      inter_sets := sorry : Filter α } →
  s ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
        Filter α } ⊓
      { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
        Filter α }
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s : Set α
mk :
  s ∈
      { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯,
        inter_sets := ⋯ : Filter α } →
    s ∈
      { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α } ⊓
        { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α }
⊢ s ∈
      { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯,
        inter_sets := ⋯ : Filter α } ↔
    s ∈
      { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α } ⊓
        { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
          Filter α }
have mk₁ : s ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
        Filter α } ⊓
      { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry :
        Filter α } →
  s ∈
    { sets := {a : Set α | ∃ (a_1 : σ_1) (b : σ_2), F.f a_1 ∩ F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry,
      inter_sets := sorry : Filter α }
Tactic: constructor
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s_1 : Set α
s : σ_1
t : σ_2
h : F.f s ∩ F_1.f t ⊆ s_1
⊢ s_1 ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
        Filter α } ⊓
      { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
        Filter α }
have apply : F.f s ∈
  { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry : Filter α }
Tactic: apply mem_inf_of_inter _ _ h
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s_1 : Set α
s : σ_1
t : σ_2
h : F.f s ∩ F_1.f t ⊆ s_1
apply :
  F.f s ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ : Filter α }
⊢ s_1 ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
        Filter α } ⊓
      { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
        Filter α }
have apply₁ : F_1.f t ∈
  { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry : Filter α }
Tactic: apply mem_inf_of_inter _ _ h
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s_1 : Set α
s : σ_1
t : σ_2
h : F.f s ∩ F_1.f t ⊆ s_1
w : σ_1
⊢ F.f s ∈
    { sets := {a : Set α | ∃ (b : σ_1), F.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ : Filter α }
have h₁ : F.f s ⊆ F.f s
Tactic: use s
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
σ_1 : Type u_5
F : CFilter.{u_1, u_5} (Set α) σ_1
σ_2 : Type u_6
F_1 : CFilter.{u_1, u_6} (Set α) σ_2
s_1 : Set α
s : σ_1
t : σ_2
h : F.f s ∩ F_1.f t ⊆ s_1
w : σ_2
⊢ F_1.f t ∈
    { sets := {a : Set α | ∃ (b : σ_2), F_1.f b ⊆ a}, univ_sets := ⋯, sets_of_superset := ⋯, inter_sets := ⋯ :
      Filter α }
have h₁ : F_1.f t ⊆ F_1.f t
Tactic: use t
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f : Filter α
m : α → Filter β
F : Realizer.{u_1, u_5} f
G : (i : α) → Realizer.{u_2, u_6} (m i)
x x_1 : (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)
x_2 : β
⊢ Membership.mem (γ := Set β)
      (match (motive := (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → Set β)
          match (motive :=
            (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)) x with
          | ⟨a, f_1⟩ =>
            match (motive :=
              (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)) x_1 with
            | ⟨b, f'⟩ =>
              Sigma.mk (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) (F.F.inf a b)
                fun (i : α) (h : i ∈ F.F.f (F.F.inf a b)) ↦ (G i).F.inf (f_1 i ⋯) (f' i ⋯) with
        | ⟨s, f_1⟩ => ⋃ (i : α), ⋃ (h : i ∈ F.F.f s), (G i).F.f (f_1 i h) :
        Set β)
      x_2 →
    Membership.mem (γ := Set β)
      (match (motive := (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → Set β) x with
        | ⟨s, f_1⟩ => ⋃ (i : α), ⋃ (h : i ∈ F.F.f s), (G i).F.f (f_1 i h) :
        Set β)
      x_2
have simp : ∀ (x_3 : α)
  (x_4 :
    x_3 ∈
      F.F.f
        (F.F.inf (x.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))
          (x_1.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))))
  (h :
    x_2 ∈
      (G x_3).F.f
        ((G x_3).F.inf (x.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) x_3 sorry)
          (x_1.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) x_3 sorry))),
  ∃ (i : α) (i_1 : i ∈ F.F.f (x.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))),
    x_2 ∈ (G i).F.f (x.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) i i_1)
Tactic: simp only [mem_iUnion, forall_exists_index]
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
f : Filter α
m : α → Filter β
F : Realizer.{u_1, u_5} f
G : (i : α) → Realizer.{u_2, u_6} (m i)
x x_1 : (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)
x_2 : β
⊢ Membership.mem (γ := Set β)
      (match (motive := (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → Set β)
          match (motive :=
            (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)) x with
          | ⟨a, f_1⟩ =>
            match (motive :=
              (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ)) x_1 with
            | ⟨b, f'⟩ =>
              Sigma.mk (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) (F.F.inf a b)
                fun (i : α) (h : i ∈ F.F.f (F.F.inf a b)) ↦ (G i).F.inf (f_1 i ⋯) (f' i ⋯) with
        | ⟨s, f_1⟩ => ⋃ (i : α), ⋃ (h : i ∈ F.F.f s), (G i).F.f (f_1 i h) :
        Set β)
      x_2 →
    Membership.mem (γ := Set β)
      (match (motive := (s : F.σ) × ((i : α) → i ∈ F.F.f s → (G i).σ) → Set β) x_1 with
        | ⟨s, f_1⟩ => ⋃ (i : α), ⋃ (h : i ∈ F.F.f s), (G i).F.f (f_1 i h) :
        Set β)
      x_2
have simp : ∀ (x_3 : α)
  (x_4 :
    x_3 ∈
      F.F.f
        (F.F.inf (x.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))
          (x_1.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))))
  (h :
    x_2 ∈
      (G x_3).F.f
        ((G x_3).F.inf (x.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) x_3 sorry)
          (x_1.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) x_3 sorry))),
  ∃ (i : α) (i_1 : i ∈ F.F.f (x_1.fst (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ))),
    x_2 ∈ (G i).F.f (x_1.snd (β := fun (s : F.σ) ↦ (i : α) → i ∈ F.F.f s → (G i).σ) i i_1)
Tactic: simp only [mem_iUnion, forall_exists_index]
-- Lean 4 proof step from Mathlib.Data.Analysis.Filter (goal)
Goal:
α : Type u_1
f : Filter α
F : Realizer.{u_1, u_5} f
⊢ f ≠ ⊥ ↔ ∀ (a : F.σ), (F.F.f a).Nonempty
have rw : (∃ (x : F.σ), ¬(F.F.f x).Nonempty) ↔
  ∀ (b : σ.{u_1, 0} (α := α) Realizer.bot), ∃ (a : F.σ), F.F.f a ≤ CFilter.f (Realizer.F Realizer.bot) b
Tactic: rw [not_iff_comm, ← le_bot_iff, F.le_iff Realizer.bot, not_forall]