awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
s : Set X
hs : IsOpen[inst] s
hne : s.Nonempty
h : IsMeagre s
⊢ sᶜ ∈ residual X
have rwa : ∀ (h : sᶜ ∈ residual X), sᶜ ∈ residual X
Tactic: rwa [IsMeagre] at h
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
s : Set X
hs : IsOpen[inst] s
hne : s.Nonempty
h : IsMeagre s
⊢ False
have obtain : ∀ (x : X) (hx : x ∈ s) (hxc : x ∈ sᶜ), False
Tactic: obtain ⟨x, hx, hxc⟩ :=
    (dense_of_mem_residual (by rwa [IsMeagre] at h)).inter_open_nonempty s hs hne
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
α : Type u_2
inst : TopologicalSpace X
inst_1 : BaireSpace X
S : Set α
f : (x : α) → x ∈ S → Set X
ho : ∀ (s : α) (H : s ∈ S), IsGδ (f s H)
hS : S.Countable
hd : ∀ (s : α) (H : s ∈ S), Dense (f s H)
⊢ Dense.{u_1} (X := X) (⋂ (s : α), ⋂ (h : s ∈ S), f s h)
have rw : Dense.{u_1} (X := X) (⋂ (x : (↑S : Type u_2)), f (↑x : α) sorry)
Tactic: rw [biInter_eq_iInter]
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (hypothesis)
Goal:
X : Type u_1
α : Type u_2
inst : TopologicalSpace X
inst_1 : BaireSpace X
S : Set α
f : (x : α) → x ∈ S → Set X
ho : ∀ (s : α) (H : s ∈ S), IsGδ (f s H)
hS : S.Countable
hd : ∀ (s : α) (H : s ∈ S), Dense (f s H)
⊢ Dense.{u_1} (X := X) (⋂ (x : (↑S : Type u_2)), f (↑x : α) ⋯)
have this : Countable.{u_2 + 1} (↑S : Type u_2)
Tactic: haveI := hS.to_subtype
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
s t : Set X
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ Dense (s ∩ t)
have rw : Dense.{u_1} (X := X) (⋂ (b : Bool), bif b then s else t)
Tactic: rw [inter_eq_iInter]
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
s t : Set X
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ Dense.{u_1} (X := X) (⋂ (b : Bool), bif b then s else t)
have ho : ∀ (s_1 : Bool), IsGδ (bif s_1 then s else t)
Tactic: apply dense_iInter_of_Gδ
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
s t : Set X
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
ho : ∀ (s_1 : Bool), IsGδ (bif s_1 then s else t)
⊢ Dense.{u_1} (X := X) (⋂ (b : Bool), bif b then s else t)
have hd : ∀ (s_1 : Bool), Dense (bif s_1 then s else t)
Tactic: apply dense_iInter_of_Gδ
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (hypothesis)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
⊢ Dense.{u_1} (X := X) (⋃ (i : ι), interior (f i))
have hgo : ∀ (i : ι), IsOpen[inst] (g i)
Tactic: have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
⊢ Dense.{u_1} (X := X) (⋂ (i : ι), g i)
have refine : ∀ (i : ι) (x : X), x ∈ closure[inst] (g i)
Tactic: refine dense_iInter_of_isOpen hgo fun i x => ?_
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
i : ι
x : X
⊢ x ∈ closure[inst] (g i)
have rw : Membership.mem (γ := Set X) ∅ᶜ x
Tactic: rw [closure_compl, interior_frontier (hc _)]
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (hypothesis)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
⊢ Dense.{u_1} (X := X) (⋃ (i : ι), interior (f i))
have hgd : Dense.{u_1} (X := X) (⋂ (i : ι), g i)
Tactic: have hgd : Dense (⋂ i, g i) := by
    refine dense_iInter_of_isOpen hgo fun i x => ?_
    rw [closure_compl, interior_frontier (hc _)]
    exact id
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
hgd : Dense.{u_1} (X := X) (⋂ (i : ι), g i)
⊢ Dense.{u_1} (X := X) (⋃ (i : ι), interior (f i))
have refine : s ∩ ⋂ (i : ι), g i ⊆ ⋃ (i : ι), interior (f i)
Tactic: refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i)) hgd).mono ?_
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (hypothesis)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
hgd : Dense.{u_1} (X := X) (⋂ (i : ι), g i)
x : X
hxs : x ∈ s
hxg : Membership.mem (γ := Set X) (⋂ (i : ι), g i) x
⊢ Membership.mem (γ := Set X) (⋃ (i : ι), interior (f i)) x
have hxg₁ : ∀ (i : ι), x ∈ g i
Tactic: rw [mem_iInter] at hxg
Cleared hypotheses: hxg
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
ι : Sort u_3
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Countable ι
s : Set X
hs : IsGδ s
hd : Dense s
f : ι → Set X
hc : ∀ (i : ι), IsClosed[inst] (f i)
hU : s ⊆ ⋃ (i : ι), f i
g : (i : ι) → Set X := fun (i : ι) ↦ (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen[inst] (g i)
hgd : Dense.{u_1} (X := X) (⋂ (i : ι), g i)
x : X
hxs : x ∈ s
hxg : ∀ (i : ι), x ∈ g i
⊢ Membership.mem (γ := Set X) (⋃ (i : ι), interior (f i)) x
have rcases : ∀ (i : ι) (hi : x ∈ f i), Membership.mem (γ := Set X) (⋃ (i : ι), interior (f i)) x
Tactic: rcases mem_iUnion.1 (hU hxs) with ⟨i, hi⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (hypothesis)
Goal:
X : Type u_1
α : Type u_2
inst : TopologicalSpace X
inst_1 : BaireSpace X
t : Set α
s : Set X
hs : IsGδ s
hd : Dense s
ht : t.Countable
f : α → Set X
hc : ∀ i ∈ t, IsClosed[inst] (f i)
hU : s ⊆ ⋃ i ∈ t, f i
⊢ Dense.{u_1} (X := X) (⋃ i ∈ t, interior (f i))
have this : Countable.{u_2 + 1} (↑t : Type u_2)
Tactic: haveI := ht.to_subtype
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
α : Type u_2
inst : TopologicalSpace X
inst_1 : BaireSpace X
t : Set α
s : Set X
hs : IsGδ s
hd : Dense s
ht : t.Countable
f : α → Set X
hc : ∀ i ∈ t, IsClosed[inst] (f i)
hU : s ⊆ ⋃ i ∈ t, f i
this : Countable.{u_2 + 1} (↑t : Type u_2)
⊢ Dense.{u_1} (X := X) (⋃ i ∈ t, interior (f i))
have simp : ∀ (hc : ∀ (x : (↑t : Type u_2)), IsClosed[inst] (f (↑x : α))) (hU : s ⊆ ⋃ (x : (↑t : Type u_2)), f (↑x : α)),
  Dense.{u_1} (X := X) (⋃ (x : (↑t : Type u_2)), interior (f (↑x : α)))
Tactic: simp only [biUnion_eq_iUnion, SetCoe.forall'] at *
Cleared hypotheses: hc, hU
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Nonempty X
s : Set X
hs : IsGδ s
hd : Dense s
h : IsMeagre s
⊢ False
have rcases : ∀ (t : Set X) (hts : t ⊆ sᶜ) (htG : IsGδ t) (hd' : Dense t), False
Tactic: rcases mem_residual.1 h with ⟨t, hts, htG, hd'⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Nonempty X
s : Set X
hs : IsGδ s
hd : Dense s
h : IsMeagre s
t : Set X
hts : t ⊆ sᶜ
htG : IsGδ t
hd' : Dense t
⊢ False
have rcases : ∀ (x : X) (hx₁ : x ∈ s) (hx₂ : x ∈ t), False
Tactic: rcases (hd.inter_of_Gδ hs htG hd').nonempty with ⟨x, hx₁, hx₂⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.Lemmas (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
inst_1 : BaireSpace X
inst_2 : Nonempty X
s : Set X
hs : s ∈ residual X
⊢ ¬IsMeagre s
have rcases : ∀ (t : Set X) (ht_sub : t ⊆ s) (htGδ : IsGδ t) (ht_dense : Dense t), ¬IsMeagre s
Tactic: rcases (mem_residual (X := X)).1 hs with ⟨t, ht_sub, htGδ, ht_dense⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
⊢ BaireSpace X
have baire_property : ∀ (f : ℕ → Set X), (∀ (n : ℕ), IsOpen (f n)) → (∀ (n : ℕ), Dense (f n)) → Dense.{u_1} (X := X) (⋂ (n : ℕ), f n)
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
⊢ Dense.{u_1} (X := X) (⋂ (n : ℕ), f n)
have baire_property : ∀ (U : Set X), IsOpen U → U.Nonempty → (U ∩ ⋂ (n : ℕ), f n).Nonempty
Tactic: rw [dense_iff_inter_open]
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
⊢ ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
have rcases : ∀ (K₀ : PositiveCompacts X) (hK₀ : closure (↑K₀ : Set X) ⊆ U),
  ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
Tactic: rcases U_open.exists_positiveCompacts_closure_subset U_nonempty with ⟨K₀, hK₀⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
⊢ ∀ (n : ℕ) (K : PositiveCompacts X), ∃ (K' : PositiveCompacts X), closure (↑K' : Set X) ⊆ f n ∩ interior (↑K : Set X)
have refine : ∀ (n : ℕ) (K : PositiveCompacts X), (f n ∩ interior (↑K : Set X)).Nonempty
Tactic: refine fun n K ↦ ((ho n).inter isOpen_interior).exists_positiveCompacts_closure_subset ?_
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
n : ℕ
K : PositiveCompacts X
⊢ (f n ∩ interior (↑K : Set X)).Nonempty
have rw : (interior (↑K : Set X) ∩ f n).Nonempty
Tactic: rw [inter_comm]
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (hypothesis)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
⊢ ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
have this : ∀ (n : ℕ) (K : PositiveCompacts X), ∃ (K' : PositiveCompacts X), closure (↑K' : Set X) ⊆ f n ∩ interior (↑K : Set X)
Tactic: have : ∀ (n) (K : PositiveCompacts X),
        ∃ K' : PositiveCompacts X, closure ↑K' ⊆ f n ∩ interior K := by
      refine fun n K ↦ ((ho n).inter isOpen_interior).exists_positiveCompacts_closure_subset ?_
      rw [inter_comm]
      exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
this :
  ∀ (n : ℕ) (K : PositiveCompacts X), ∃ (K' : PositiveCompacts X), closure (↑K' : Set X) ⊆ f n ∩ interior (↑K : Set X)
⊢ ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
have choose : ∀ (K_next : (n : ℕ) → (K : PositiveCompacts X) → PositiveCompacts X)
  (hK_next : ∀ (n : ℕ) (K : PositiveCompacts X), closure (↑(K_next n K) : Set X) ⊆ f n ∩ interior (↑K : Set X)),
  ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
Tactic: choose K_next hK_next using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
K_next : (n : ℕ) → (K : PositiveCompacts X) → PositiveCompacts X
hK_next : ∀ (n : ℕ) (K : PositiveCompacts X), closure (↑(K_next n K) : Set X) ⊆ f n ∩ interior (↑K : Set X)
⊢ ∃ (K : ℕ → PositiveCompacts X),
    (∀ (n : ℕ), K (n + 1) ≤ K n) ∧ (∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) ∧ closure (↑(K 0) : Set X) ⊆ U
have h : (∀ (n : ℕ),
    LE.le.{u_1} (α := PositiveCompacts X)
      ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) (n + 1))
      ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) n)) ∧
  (∀ (n : ℕ),
      closure (↑((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) (n + 1)) : Set X) ⊆
        f n) ∧
    closure (↑((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) 0) : Set X) ⊆ U
Tactic: use Nat.rec K₀ K_next
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K₀ : PositiveCompacts X
hK₀ : closure (↑K₀ : Set X) ⊆ U
K_next : (n : ℕ) → (K : PositiveCompacts X) → PositiveCompacts X
hK_next : ∀ (n : ℕ) (K : PositiveCompacts X), closure (↑(K_next n K) : Set X) ⊆ f n ∩ interior (↑K : Set X)
⊢ (∀ (n : ℕ),
      LE.le.{u_1} (α := PositiveCompacts X)
        ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) (n + 1))
        ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) n)) ∧
    (∀ (n : ℕ),
        closure (↑((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) (n + 1)) : Set X) ⊆
          f n) ∧
      closure (↑((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) 0) : Set X) ⊆ U
have h : ∀ (n : ℕ),
  LE.le.{u_1} (α := PositiveCompacts X)
    ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) (n + 1))
    ((fun (t : ℕ) ↦ Nat.rec (motive := fun (x : ℕ) ↦ PositiveCompacts X) K₀ K_next t) n)
Tactic: refine ⟨fun n ↦ ?_, fun n ↦ (hK_next n _).trans inter_subset_left, hK₀⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
⊢ (U ∩ ⋂ (n : ℕ), f n).Nonempty
have baire_property : ∀ (K : ℕ → PositiveCompacts X) (hK_anti : ∀ (n : ℕ), K (n + 1) ≤ K n)
  (hKf : ∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n) (hKU : closure (↑(K 0) : Set X) ⊆ U),
  (U ∩ ⋂ (n : ℕ), f n).Nonempty
Tactic: obtain ⟨K, hK_anti, hKf, hKU⟩ : ∃ K : ℕ → PositiveCompacts X,
      (∀ n, K (n + 1) ≤ K n) ∧ (∀ n, closure ↑(K (n + 1)) ⊆ f n) ∧ closure ↑(K 0) ⊆ U := by
    rcases U_open.exists_positiveCompacts_closure_subset U_nonempty with ⟨K₀, hK₀⟩
    have : ∀ (n) (K : PositiveCompacts X),
        ∃ K' : PositiveCompacts X, closure ↑K' ⊆ f n ∩ interior K := by
      refine fun n K ↦ ((ho n).inter isOpen_interior).exists_positiveCompacts_closure_subset ?_
      rw [inter_comm]
      exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty
    choose K_next hK_next using this
    -- The next two lines are faster than a single `refine`.
    use Nat.rec K₀ K_next
    refine ⟨fun n ↦ ?_, fun n ↦ (hK_next n _).trans inter_subset_left, hK₀⟩
    exact subset_closure.trans <| (hK_next _ _).trans <|
      inter_subset_right.trans interior_subset
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K : ℕ → PositiveCompacts X
hK_anti : ∀ (n : ℕ), K (n + 1) ≤ K n
hKf : ∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n
hKU : closure (↑(K 0) : Set X) ⊆ U
x : X
hx : Membership.mem (γ := Set X) (⋂ (n : ℕ), closure (↑(K n) : Set X)) x
⊢ x ∈ U ∩ ⋂ (n : ℕ), f n
have simp : ∀ (hx : ∀ (i : ℕ), x ∈ closure (↑(K i) : Set X)), x ∈ U ∧ ∀ (i : ℕ), x ∈ f i
Tactic: simp only [mem_iInter, mem_inter_iff] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (hypothesis)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K : ℕ → PositiveCompacts X
hK_anti : ∀ (n : ℕ), K (n + 1) ≤ K n
hKf : ∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n
hKU : closure (↑(K 0) : Set X) ⊆ U
⊢ (U ∩ ⋂ (n : ℕ), f n).Nonempty
have hK_subset : ⋂ (n : ℕ), closure (↑(K n) : Set X) ⊆ U ∩ ⋂ (n : ℕ), f n
Tactic: have hK_subset : (⋂ n, closure (K n) : Set X) ⊆ U ∩ ⋂ n, f n := fun x hx ↦ by
    simp only [mem_iInter, mem_inter_iff] at hx ⊢
    exact ⟨hKU <| hx 0, fun n ↦ hKf n <| hx (n + 1)⟩
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (hypothesis)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set X
U_open : IsOpen U
U_nonempty : U.Nonempty
K : ℕ → PositiveCompacts X
hK_anti : ∀ (n : ℕ), K (n + 1) ≤ K n
hKf : ∀ (n : ℕ), closure (↑(K (n + 1)) : Set X) ⊆ f n
hKU : closure (↑(K 0) : Set X) ⊆ U
hK_subset : ⋂ (n : ℕ), closure (↑(K n) : Set X) ⊆ U ∩ ⋂ (n : ℕ), f n
⊢ (U ∩ ⋂ (n : ℕ), f n).Nonempty
have hK_nonempty : Set.Nonempty.{u_1} (α := X) (⋂ (n : ℕ), closure (↑(K n) : Set X))
Tactic: have hK_nonempty : (⋂ n, closure (K n) : Set X).Nonempty :=
    IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed _
      (fun n => closure_mono <| hK_anti n) (fun n => (K n).nonempty.closure)
      (K 0).isCompact.closure fun n => isClosed_closure
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
hG : IsGδ s
⊢ BaireSpace.{u_1} (↑(closure s) : Type u_1)
have convert_3 : R1Space.{u_1} (↑(closure s) : Type u_1)
Tactic: convert! BaireSpace.of_t2Space_locallyCompactSpace using 1
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
hG : IsGδ s
convert_3 : R1Space.{u_1} (↑(closure s) : Type u_1)
⊢ BaireSpace.{u_1} (↑(closure s) : Type u_1)
have convert_4 : LocallyCompactSpace.{u_1} (↑(closure s) : Type u_1)
Tactic: convert! BaireSpace.of_t2Space_locallyCompactSpace using 1
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (hypothesis)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
hG : IsGδ s
⊢ BaireSpace.{u_1} (↑s : Type u_1)
have this : BaireSpace.{u_1} (↑(closure s) : Type u_1)
Tactic: have : BaireSpace (closure s) := by
    convert! BaireSpace.of_t2Space_locallyCompactSpace using 1
    · infer_instance
    · exact isClosed_closure.locallyCompactSpace
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (hypothesis)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
hG : IsGδ s
this : BaireSpace.{u_1} (↑(closure s) : Type u_1)
⊢ BaireSpace.{u_1} (↑s : Type u_1)
have this₁ : BaireSpace.{u_1} (↑(Subtype.val (p := fun (x : X) ↦ x ∈ closure s) ⁻¹' s) : Type u_1)
Tactic: have : BaireSpace ((↑) ⁻¹' s : Set (closure s)) :=
    (isGδ_induced continuous_subtype_val hG).baireSpace_of_dense
    (by simp [Subtype.dense_iff, inter_eq_right.mpr subset_closure])
-- Lean 4 proof step from Mathlib.Topology.Baire.LocallyCompactRegular (goal)
Goal:
X : Type u_1
inst : TopologicalSpace X
s : Set X
inst_1 : R1Space X
inst_2 : LocallyCompactSpace X
hG : IsGδ s
this_1 : BaireSpace.{u_1} (↑(closure s) : Type u_1)
this : BaireSpace.{u_1} (↑(Subtype.val (p := fun (x : X) ↦ x ∈ closure s) ⁻¹' s) : Type u_1)
⊢ BaireSpace.{u_1} (↑s : Type u_1)
have have : ∀ (h_homeo : (↑(Subtype.val (p := fun (x : X) ↦ x ∈ closure s) ⁻¹' s) : Type u_1) ≃ₜ (↑s : Type u_1)),
  BaireSpace.{u_1} (↑s : Type u_1)
Tactic: have h_homeo : Homeomorph ((↑) ⁻¹' s : Set (closure s)) s := ⟨⟨fun x => ⟨x, x.2⟩,
    fun x => ⟨⟨x, subset_closure x.2⟩, x.2⟩, by grind, by grind⟩, by fun_prop, by fun_prop⟩
-- Lean 4 proof step from Mathlib.Topology.Bases (goal)
Goal:
α : Type u
t : TopologicalSpace α
s : Set (Set α)
hs : t = generateFrom s
⊢ IsTopologicalBasis.{u} (α := α) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s})
have subst : IsTopologicalBasis.{u} (α := α) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s})
Tactic: subst t
Cleared hypotheses: t, hs
-- Lean 4 proof step from Mathlib.Topology.Bases (goal)
Goal:
α : Type u
s : Set (Set α)
this : TopologicalSpace α := generateFrom s
⊢ IsTopologicalBasis.{u} (α := α) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s})
have refine_1 : ∀ (t₁ : Set α),
  Membership.mem (γ := Set (Set α)) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s}) t₁ →
    ∀ (t₂ : Set α),
      Membership.mem (γ := Set (Set α)) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s}) t₂ →
        ∀ x ∈ t₁ ∩ t₂,
          ∃ (t₃ : Set α),
            Membership.mem (γ := Set (Set α)) ((fun (f : Set (Set α)) ↦ ⋂₀ f) '' {f : Set (Set α) | f.Finite ∧ f ⊆ s})
                t₃ ∧
              x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
Tactic: refine ⟨?_, ?_, le_antisymm (le_generateFrom ?_) <| generateFrom_anti fun t ht => ?_⟩