awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
x : X
hx : x ∈ s
j : ι
hji : j ≠ i
hj : x ∈ v.toFun (u := u) (s := s) (p := ⊤) j
⊢ ∃ (i_1 : ι), x ∈ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
have h : x ∈ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi j
Tactic: use j
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
x : X
hx : x ∈ s
h : ∀ (j : ι) (_ : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := ⊤) j
⊢ ∃ (i_1 : ι), x ∈ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
have h₁ : x ∈ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i
Tactic: use i
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
x : X
hx : x ∈ s
h : ∀ (j : ι) (_ : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := ⊤) j
⊢ x ∈ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i
have h₁ : x ∈ vi
Tactic: rw [update_self]
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
vi : Set X
ovi : IsOpen vi
j : ι
hi : j ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j_1 : ι), ⋂ (_ : j_1 ≠ j), (v.toFun (u := u) (s := s) (p := ⊤) j_1)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) j
C : IsClosed (s ∩ ⋂ (j_1 : ι), ⋂ (_ : j_1 ≠ j), (v.toFun (u := u) (s := s) (p := ⊤) j_1)ᶜ)
hvi : s ∩ ⋂ (j_1 : ι), ⋂ (_ : j_1 ≠ j), (v.toFun (u := u) (s := s) (p := ⊤) j_1)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) j
⊢ closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) j vi j) ⊆ u j
have refine_3 : closure vi ⊆ u j
Tactic: rwa [update_self, ← v.apply_eq hi]
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
j : ι
hj : j ∈ v.carrier (u := u) (s := s) (p := ⊤)
⊢ closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi j) ⊆ u j
have refine_3 : closure (v.toFun (u := u) (s := s) (p := ⊤) j) ⊆ u j
Tactic: rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (hypothesis)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
j : ι
hj : j ∉ insert i (v.carrier (u := u) (s := s) (p := ⊤))
⊢ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi j = u j
have hj₁ : ¬j = i ∧ j ∉ v.carrier (u := u) (s := s) (p := ⊤)
Tactic: rw [mem_insert_iff, not_or] at hj
Cleared hypotheses: hj
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
j : ι
hj : ¬j = i ∧ j ∉ v.carrier (u := u) (s := s) (p := ⊤)
⊢ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi j = u j
have refine_5 : v.toFun (u := u) (s := s) (p := ⊤) j = u j
Tactic: rw [update_of_ne hj.1, v.apply_eq hj.2]
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
j : ι
hj : ¬j = i ∧ j ∉ v.carrier (u := u) (s := s) (p := ⊤)
refine_5 : v.toFun (u := u) (s := s) (p := ⊤) j = u j
⊢ update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi j = u j
have refine_5₁ : u j = u j
Tactic: rw [update_of_ne hj.1, v.apply_eq hj.2]
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
⊢ (fun (v₁ v₂ : PartialRefinement u s ⊤) ↦
      v₁.carrier (u := u) (s := s) (p := ⊤) ⊆ v₂.carrier (u := u) (s := s) (p := ⊤) ∧
        ∀ i ∈ v₁.carrier (u := u) (s := s) (p := ⊤),
          v₁.toFun (u := u) (s := s) (p := ⊤) i = v₂.toFun (u := u) (s := s) (p := ⊤) i)
    v
    (mk (u := u) (s := s) (p := ⊤) (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi)
      (insert i (v.carrier (u := u) (s := s) (p := ⊤))) ⋯ ⋯ ⋯ ⋯ ⋯)
have refine_6 : ∀ (j : ι) (hj : j ∈ v.carrier (u := u) (s := s) (p := ⊤)),
  v.toFun (u := u) (s := s) (p := ⊤) j =
    toFun (u := u) (s := s) (p := ⊤)
      (mk (u := u) (s := s) (p := ⊤) (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi)
        (insert i (v.carrier (u := u) (s := s) (p := ⊤))) sorry sorry sorry sorry sorry)
      j
Tactic: refine ⟨subset_insert _ _, fun j hj => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_2 : s ⊆ ⋃ (i_1 : ι), update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
refine_2 : s ⊆ ⋃ (i_1 : ι), update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_3 : ∀ {i_1 : ι},
  i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
    closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1) ⊆ u i_1
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
refine_2 : s ⊆ ⋃ (i_1 : ι), update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
refine_3 :
  ∀ {i_1 : ι},
    i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
      closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1) ⊆ u i_1
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_4 : ∀ {i_1 : ι} (hi : i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤))),
  ⊤ (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
refine_2 : s ⊆ ⋃ (i_1 : ι), update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
refine_3 :
  ∀ {i_1 : ι},
    i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
      closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1) ⊆ u i_1
refine_4 :
  ∀ {i_1 : ι} (hi : i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤))),
    ⊤ (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_5 : ∀ {i_1 : ι},
  i_1 ∉ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
    update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1 = u i_1
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
I : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
C : IsClosed (s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ)
vi : Set X
ovi : IsOpen vi
hvi : s ∩ ⋂ (j : ι), ⋂ (_ : j ≠ i), (v.toFun (u := u) (s := s) (p := ⊤) j)ᶜ ⊆ vi
cvi : closure vi ⊆ v.toFun (u := u) (s := s) (p := ⊤) i
refine_1 : ∀ (i_1 : ι), IsOpen (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
refine_2 : s ⊆ ⋃ (i_1 : ι), update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1
refine_3 :
  ∀ {i_1 : ι},
    i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
      closure (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1) ⊆ u i_1
refine_4 :
  ∀ {i_1 : ι} (hi : i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := ⊤))),
    ⊤ (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1)
refine_5 :
  ∀ {i_1 : ι},
    i_1 ∉ insert i (v.carrier (u := u) (s := s) (p := ⊤)) →
      update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi i_1 = u i_1
⊢ ∃ (v' : PartialRefinement u s ⊤), v < v'
have refine_6 : (fun (v₁ v₂ : PartialRefinement u s ⊤) ↦
    v₁.carrier (u := u) (s := s) (p := ⊤) ⊆ v₂.carrier (u := u) (s := s) (p := ⊤) ∧
      ∀ i ∈ v₁.carrier (u := u) (s := s) (p := ⊤),
        v₁.toFun (u := u) (s := s) (p := ⊤) i = v₂.toFun (u := u) (s := s) (p := ⊤) i)
  v
  (mk (u := u) (s := s) (p := ⊤) (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := ⊤)) i vi)
    (insert i (v.carrier (u := u) (s := s) (p := ⊤))) sorry sorry sorry sorry sorry)
Tactic: classical
  refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ?_, ?_⟩
  · intro j
    rcases eq_or_ne j i with (rfl | hne) <;> simp [*, v.isOpen]
  · refine fun x hx => mem_iUnion.2 ?_
    by_cases! h : ∃ j ≠ i, x ∈ v j
    · rcases h with ⟨j, hji, hj⟩
      use j
      rwa [update_of_ne hji]
    · use i
      rw [update_self]
      exact hvi ⟨hx, mem_biInter h⟩
  · rintro j (rfl | hj)
    · rwa [update_self, ← v.apply_eq hi]
    · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
      exact v.closure_subset hj
  · exact fun _ => trivial
  · intro j hj
    rw [mem_insert_iff, not_or] at hj
    rw [update_of_ne hj.1, v.apply_eq hj.2]
  · refine ⟨subset_insert _ _, fun j hj => ?_⟩
    exact (update_of_ne (ne_of_mem_of_not_mem hj hi) _ _).symm
  · exact fun hle => hi (hle.1 <| mem_insert _ _)
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (hypothesis)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, Set.Finite.{u_1} (α := ι) {i : ι | x ∈ u i}
us : s ⊆ ⋃ (i : ι), u i
⊢ ∃ (v : ι → Set X), s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
have this : Nonempty (PartialRefinement u s ⊤)
Tactic: haveI : Nonempty (PartialRefinement u s ⊤) :=
    ⟨⟨u, ∅, uo, us, False.elim, False.elim, fun _ => rfl⟩⟩
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (hypothesis)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, Set.Finite.{u_1} (α := ι) {i : ι | x ∈ u i}
us : s ⊆ ⋃ (i : ι), u i
this : Nonempty (PartialRefinement u s ⊤)
⊢ ∃ (v : ι → Set X), s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
have this₁ : ∀ (c : Set (PartialRefinement u s ⊤)),
  IsChain (fun (x1 x2 : PartialRefinement u s ⊤) ↦ x1 ≤ x2) c →
    c.Nonempty → ∃ (ub : PartialRefinement u s ⊤), ∀ v ∈ c, v ≤ ub
Tactic: have : ∀ c : Set (PartialRefinement u s ⊤),
      IsChain (· ≤ ·) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub :=
    fun c hc ne => ⟨.chainSup c hc ne uf us, fun v hv => PartialRefinement.le_chainSup _ _ _ _ hv⟩
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, Set.Finite.{u_1} (α := ι) {i : ι | x ∈ u i}
us : s ⊆ ⋃ (i : ι), u i
this_1 : Nonempty (PartialRefinement u s ⊤)
this :
  ∀ (c : Set (PartialRefinement u s ⊤)),
    IsChain (fun (x1 x2 : PartialRefinement u s ⊤) ↦ x1 ≤ x2) c →
      c.Nonempty → ∃ (ub : PartialRefinement u s ⊤), ∀ v ∈ c, v ≤ ub
⊢ ∃ (v : ι → Set X), s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
have rcases : ∀ (v : PartialRefinement u s ⊤) (hv : IsMax v),
  ∃ (v : ι → Set X), s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
Tactic: rcases zorn_le_nonempty this with ⟨v, hv⟩
-- Lean 4 proof step from Mathlib.Topology.ShrinkingLemma (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : TopologicalSpace X
u : ι → Set X
s : Set X
inst_1 : NormalSpace X
hs : IsClosed s
uo : ∀ (i : ι), IsOpen (u i)
uf : ∀ x ∈ s, Set.Finite.{u_1} (α := ι) {i : ι | x ∈ u i}
us : s ⊆ ⋃ (i : ι), u i
this_1 : Nonempty (PartialRefinement u s ⊤)
this :
  ∀ (c : Set (PartialRefinement u s ⊤)),
    IsChain (fun (x1 x2 : PartialRefinement u s ⊤) ↦ x1 ≤ x2) c →
      c.Nonempty → ∃ (ub : PartialRefinement u s ⊤), ∀ v ∈ c, v ≤ ub
v : PartialRefinement u s ⊤
hv : IsMax v
⊢ ∃ (v : ι → Set X), s ⊆ iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
have suffices : ∀ (i : ι), i ∈ v.carrier (u := u) (s := s) (p := ⊤)
Tactic: suffices ∀ i, i ∈ v.carrier from
    ⟨v, v.subset_iUnion, fun i => v.isOpen _, fun i => v.closure_subset (this i)⟩