awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
rw rw₁ rw₂ :
  IsCompactOpenCovered (X := X) f U ↔
    ∃ (V : Opens ((i : ι) × X i)),
      IsCompact V.carrier ∧
        (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U
refine_1 :
  IsCompactOpenCovered (X := X) f U →
    ∀ (s : Set ι) (hs : s.Finite) (V : (i : ι) → i ∈ s → Opens (X i))
      (hc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier)
      (hU : ⋃ (i : ι), ⋃ (h : i ∈ s), f i '' (↑(V i h) : Set (X i)) = U),
      ∃ (V : Opens ((i : ι) × X i)),
        IsCompact V.carrier ∧
          (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
            U
⊢ IsCompactOpenCovered (X := X) f U ↔
    IsCompactOpenCovered (X := fun (x : Unit) ↦ (i : ι) × X i)
      (fun (x : Unit) (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) U
have refine_2 : (∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
        U) →
  ∀ (V : Opens ((i : ι) × X i)) (hc : IsCompact V.carrier)
    (heq :
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U),
    IsCompactOpenCovered (X := X) f U
Tactic: classical
  rw [iff_of_unique (ι := Unit)]
  refine ⟨fun ⟨s, hs, V, hc, hU⟩ ↦ ?_, fun ⟨V, hc, heq⟩ ↦ ?_⟩
  · refine ⟨⟨s.sigma fun i ↦ if h : i ∈ s then V i h else ∅, isOpen_sigma_iff.mpr ?_⟩, ?_, ?_⟩
    · intro i
      by_cases h : i ∈ s
      · simpa [h] using (V _ _).2
      · simp [h]
    · dsimp only
      exact Set.isCompact_sigma hs fun i ↦ (by simp_all)
    · aesop
  · obtain ⟨s, t, hs, hc, heq'⟩ := hc.sigma_exists_finite_sigma_eq
    have (i : ι) (hi : i ∈ s) : IsOpen (t i) := by
      rw [← Set.mk_preimage_sigma (t := t) hi]
      exact isOpen_sigma_iff.mp (heq' ▸ V.2) i
    refine ⟨s, hs, fun i hi ↦ ⟨t i, this i hi⟩, fun i _ ↦ hc i, ?_⟩
    simp_rw [coe_mk, ← heq, ← heq', Set.image_sigma_eq_iUnion, Function.comp_apply]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
⊢ IsCompactOpenCovered (X := X) f U
have rw : ∃ (V : Opens ((i : ι) × X i)),
  IsCompact V.carrier ∧
    (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U
Tactic: rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
i : κ
⊢ ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = s i
have a : (∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
        s i) ↔
  IsCompactOpenCovered (X := X) f (s i)
Tactic: convert! H i
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
i : κ
⊢ (∃ (V : Opens ((i : ι) × X i)),
      IsCompact V.carrier ∧
        (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
          s i) ↔
    IsCompactOpenCovered (X := X) f (s i)
have a : (∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
        s i) ↔
  IsCompactOpenCovered (X := fun (x : Unit) ↦ (i : ι) × X i)
    (fun (x : Unit) (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) (s i)
Tactic: rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
i : κ
a :
  (∃ (V : Opens ((i : ι) × X i)),
      IsCompact V.carrier ∧
        (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
          s i) ↔
    IsCompactOpenCovered (X := fun (x : Unit) ↦ (i : ι) × X i)
      (fun (x : Unit) (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) (s i)
⊢ (∃ (V : Opens ((i : ι) × X i)),
      IsCompact V.carrier ∧
        (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
          s i) ↔
    IsCompactOpenCovered (X := X) f (s i)
have a₁ : (∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
        s i) ↔
  ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = s i
Tactic: rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (hypothesis)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
⊢ ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U
have this : ∀ (i : κ),
  ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = s i
Tactic: have (i : κ) : ∃ (V : Opens (Σ i, X i)), IsCompact V.1 ∧ (f _ ·.snd) '' V.1 = s i := by
    convert! H i; rw [iff_isCompactOpenCovered_sigmaMk, iff_of_unique]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
κ : Type u_4
inst_1 : Finite κ
s : κ → Set S
hs : ⋃ (i : κ), s i = U
H : ∀ (i : κ), IsCompactOpenCovered (X := X) f (s i)
this :
  ∀ (i : κ),
    ∃ (V : Opens ((i : ι) × X i)),
      IsCompact V.carrier ∧
        (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' V.carrier =
          s i
⊢ ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U
have choose : ∀ (V : (i : κ) → Opens ((i : ι) × X i)) (hVeq : ∀ (i : κ), IsCompact (V i).carrier)
  (hVc :
    ∀ (i : κ),
      (fun (x : (i : ι) × X i) ↦ f (x.fst (β := fun (i : ι) ↦ X i)) (x.snd (β := fun (i : ι) ↦ X i))) '' (V i).carrier =
        s i),
  ∃ (V : Opens ((i : ι) × X i)),
    IsCompact V.carrier ∧
      (fun (p : (i : ι) × X i) ↦ f (p.fst (β := fun (i : ι) ↦ X i)) (p.snd (β := fun (i : ι) ↦ X i))) '' V.carrier = U
Tactic: choose V hVeq hVc using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (hypothesis)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
U : Set S
s : Set (Set S)
hs : ⋃ t ∈ s, t = U
hf : s.Finite
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f t
⊢ IsCompactOpenCovered (X := X) f U
have this : Finite.{u_1 + 1} (↑s : Type u_1)
Tactic: have := hf.to_subtype
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
⊢ IsCompactOpenCovered (X := X) f U
have obtain : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)), IsCompactOpenCovered (X := X) f U
Tactic: obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
t : Finset (↑s : Type u_1)
ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)
⊢ IsCompactOpenCovered (X := X) f U
have refine_1 : ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
Tactic: refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
t : Finset (↑s : Type u_1)
ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)
refine_1 :
  ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
⊢ IsCompactOpenCovered (X := X) f U
have refine_2 : (SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S))).Finite
Tactic: refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
t : Finset (↑s : Type u_1)
ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)
refine_1 :
  ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
refine_2 : (SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S))).Finite
⊢ IsCompactOpenCovered (X := X) f U
have refine_3 : ∀ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)),
  IsCompactOpenCovered (X := X) f t_1
Tactic: refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
⊢ IsCompactOpenCovered (X := X) f U
have obtain : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)), IsCompactOpenCovered (X := X) f U
Tactic: classical
  obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
  refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
  · exact subset_antisymm (fun x h ↦ by aesop) (subset_trans ht <| by simp)
  · exact Set.toFinite _
  · grind
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
obtain : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)), IsCompactOpenCovered (X := X) f U
⊢ IsCompactOpenCovered (X := X) f U
have refine_1 : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
  ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
Tactic: classical
  obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
  refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
  · exact subset_antisymm (fun x h ↦ by aesop) (subset_trans ht <| by simp)
  · exact Set.toFinite _
  · grind
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
obtain : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)), IsCompactOpenCovered (X := X) f U
refine_1 :
  ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
    ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
⊢ IsCompactOpenCovered (X := X) f U
have refine_2 : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
  (SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S))).Finite
Tactic: classical
  obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
  refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
  · exact subset_antisymm (fun x h ↦ by aesop) (subset_trans ht <| by simp)
  · exact Set.toFinite _
  · grind
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, (↑t : Set S) = U
H : ∀ t ∈ s, IsCompactOpenCovered (X := X) f (↑t : Set S)
obtain : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)), IsCompactOpenCovered (X := X) f U
refine_1 :
  ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
    ⋃ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)), t_1 = U
refine_2 :
  ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
    (SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S))).Finite
⊢ IsCompactOpenCovered (X := X) f U
have refine_3 : ∀ (t : Finset (↑s : Type u_1)) (ht : U ⊆ ⋃ i ∈ t, (↑(↑i : Opens S) : Set S)),
  ∀ t_1 ∈ SetLike.coe '' (↑(Finset.image (Subtype.val (p := fun (x : Opens S) ↦ x ∈ s)) t) : Set (Opens S)),
    IsCompactOpenCovered (X := X) f t_1
Tactic: classical
  obtain ⟨t, ht⟩ := hU.elim_finite_subcover (fun V : s ↦ V.1) (fun V ↦ V.1.2) (by simp [← hs])
  refine of_biUnion_eq_of_finite (SetLike.coe '' (t.image Subtype.val : Set (Opens S))) ?_ ?_ ?_
  · exact subset_antisymm (fun x h ↦ by aesop) (subset_trans ht <| by simp)
  · exact Set.toFinite _
  · grind
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
H : ∀ x ∈ U, ∃ t ⊆ U, x ∈ t ∧ IsOpen t ∧ IsCompactOpenCovered (X := X) f t
⊢ IsCompactOpenCovered (X := X) f U
have choose : ∀ (Us : (x : S) → x ∈ U → Set S) (hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U) (hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a)
  (hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a))
  (hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)), IsCompactOpenCovered (X := X) f U
Tactic: choose Us hU' hUx hUo hU'' using H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)
⊢ IsCompactOpenCovered (X := X) f U
have refine_1 : ⋃ (t : Opens S),
    ⋃ (_ : Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
      (↑t : Set S) =
  U
Tactic: refine of_biUnion_eq_of_isCompact hU { Us x h | (x : S) (h : x ∈ U) } ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)
refine_1 :
  ⋃ (t : Opens S),
      ⋃ (_ : Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
        (↑t : Set S) =
    U
⊢ IsCompactOpenCovered (X := X) f U
have refine_2 : ∀ (t : Opens S),
  Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t →
    IsCompactOpenCovered (X := X) f (↑t : Set S)
Tactic: refine of_biUnion_eq_of_isCompact hU { Us x h | (x : S) (h : x ∈ U) } ?_ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)
⊢ ⋃ (t : Opens S),
      ⋃ (_ : Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
        (↑t : Set S) =
    U
have refine_1 : ∀ (x : S),
  Membership.mem (γ := Set S)
      (⋃ (t : Opens S),
        ⋃ (_ :
          Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
          (↑t : Set S))
      x →
    x ∈ U
Tactic: refine subset_antisymm (fun x ↦ ?_) fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)
refine_1 :
  ∀ (x : S),
    Membership.mem (γ := Set S)
        (⋃ (t : Opens S),
          ⋃ (_ :
            Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
            (↑t : Set S))
        x →
      x ∈ U
⊢ ⋃ (t : Opens S),
      ⋃ (_ : Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
        (↑t : Set S) =
    U
have refine_1₁ : ∀ (x : S) (hx : x ∈ U),
  Membership.mem (γ := Set S)
    (⋃ (t : Opens S),
      ⋃ (_ : Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
        (↑t : Set S))
    x
Tactic: refine subset_antisymm (fun x ↦ ?_) fun x hx ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
inst_1 : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered (X := X) f (Us x a)
x : S
⊢ Membership.mem (γ := Set S)
      (⋃ (t : Opens S),
        ⋃ (_ :
          Membership.mem (γ := Set (Opens S)) {x : Opens S | ∃ (x_1 : S) (h : x_1 ∈ U), Us x_1 h = (↑x : Set S)} t),
          (↑t : Set S))
      x →
    x ∈ U
have refine_1 : ∀ (U_1 : Set S) (hU : IsOpen U_1) (x_1 : S) (x_2 : x_1 ∈ U) (h : Us x_1 sorry = U_1), x ∈ U_1 → x ∈ U
Tactic: simp [Opens.forall]
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
B : (i : ι) → Set (Opens (X i))
hB : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
hU : IsCompactOpenCovered (X := X) f U
h :
  ∃ (κ : Type (max u_2 u_3)) (_ : Finite κ) (a : κ → ι) (V : (i : κ) → Opens (X (a i))),
    (∀ (i : κ), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)) ∧
      (∀ (i : κ), IsCompact (V i).carrier) ∧ ⋃ (i : κ), f (a i) '' (↑(V i) : Set (X (a i))) = U
⊢ ∃ (n : ℕ) (a : Fin n → ι) (V : (i : Fin n) → Opens (X (a i))),
    (∀ (i : Fin n), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)) ∧
      ⋃ (i : Fin n), f (a i) '' (↑(V i) : Set (X (a i))) = U
have obtain : ∀ (κ : Type (max u_2 u_3)),
  Finite κ →
    ∀ (a : κ → ι) (V : (i : κ) → Opens (X (a i)))
      (hB : ∀ (i : κ), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i))
      (hc : ∀ (i : κ), IsCompact (V i).carrier) (hU : ⋃ (i : κ), f (a i) '' (↑(V i) : Set (X (a i))) = U),
      ∃ (n : ℕ) (a : Fin n → ι) (V : (i : Fin n) → Opens (X (a i))),
        (∀ (i : Fin n), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)) ∧
          ⋃ (i : Fin n), f (a i) '' (↑(V i) : Set (X (a i))) = U
Tactic: obtain ⟨κ, _, a, V, hB, hc, hU⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Sets.CompactOpenCovered (goal)
Goal:
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst : (i : ι) → TopologicalSpace (X i)
B : (i : ι) → Set (Opens (X i))
hB_1 : ∀ (i : ι), IsBasis (B i)
hBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier
U : Set S
hU_1 : IsCompactOpenCovered (X := X) f U
κ : Type (max u_2 u_3)
w : Finite κ
a : κ → ι
V : (i : κ) → Opens (X (a i))
hB : ∀ (i : κ), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)
hc : ∀ (i : κ), IsCompact (V i).carrier
hU : ⋃ (i : κ), f (a i) '' (↑(V i) : Set (X (a i))) = U
⊢ ∃ (n : ℕ) (a : Fin n → ι) (V : (i : Fin n) → Opens (X (a i))),
    (∀ (i : Fin n), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)) ∧
      ⋃ (i : Fin n), f (a i) '' (↑(V i) : Set (X (a i))) = U
have intro : ∀ (val : Fintype κ),
  ∃ (n : ℕ) (a : Fin n → ι) (V : (i : Fin n) → Opens (X (a i))),
    (∀ (i : Fin n), Membership.mem (γ := Set (Opens (X (a i)))) (B (a i)) (V i)) ∧
      ⋃ (i : Fin n), f (a i) '' (↑(V i) : Set (X (a i))) = U
Tactic: cases nonempty_fintype κ