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
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
i : ι
⊢ i ∈ v.carrier (u := u) (s := s) (p := ⊤)
have by_contra : ∀ (hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)), False
Tactic: by_contra 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
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
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := ⊤)
⊢ False
have rcases : ∀ (v' : PartialRefinement u s ⊤) (hlt : v < v'), False
Tactic: rcases v.exists_gt hs i hi with ⟨v', hlt⟩
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have hsi : let si := s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ;
si = s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
Tactic: set si := s ∩ (⋃ j ≠ i, v j)ᶜ with hsi
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi : si = s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have hsi₁ : si = s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
Tactic: simp only [ne_eq, compl_iUnion] at hsi
Cleared hypotheses: hsi
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
⊢ IsCompact si
have apply : IsClosed (s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ)
Tactic: apply IsCompact.of_isClosed_subset hs _ Set.inter_subset_left
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
⊢ IsOpen.{u_2} (X := X)
    (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)
have h : ∀ (i_1 : ι),
  Membership.mem (γ := Set ι) (fun (i_2 : ι) ↦ i_2 = i → False) i_1 →
    IsOpen (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)
Tactic: apply isOpen_biUnion
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
⊢ IsClosed (s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ)
have this : IsOpen.{u_2} (X := X)
  (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)
Tactic: have : IsOpen (⋃ j ≠ i, v j) := by
        apply isOpen_biUnion
        intro j _
        exact v.isOpen j
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have hsic : IsCompact si
Tactic: have hsic : IsCompact si := by
    apply IsCompact.of_isClosed_subset hs _ Set.inter_subset_left
    · have : IsOpen (⋃ j ≠ i, v j) := by
        apply isOpen_biUnion
        intro j _
        exact v.isOpen j
      exact IsClosed.inter (IsCompact.isClosed hs) (IsOpen.isClosed_compl this)
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
j : ι
hj : j ≠ i
⊢ x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
have rw : ∀
  (hx :
    x ∈
      s ∩
        ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ)
  (j : ι) (hj : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
Tactic: rw [hsi] at hx
Cleared hypotheses: hx, j, hj
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx :
  x ∈ s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
j : ι
hj : j ≠ i
⊢ x ∈ (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
have hsi' : Membership.mem (γ := Set X)
  (⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ) x
Tactic: have hsi' : x ∈ (⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ) := Set.mem_of_mem_inter_right hx
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx :
  x ∈ s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
j : ι
hj : j ≠ i
hsi' :
  Membership.mem (γ := Set X)
    (⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ) x
⊢ x ∈ (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
have hj₁ : ¬j = i
Tactic: rw [ne_eq] at hj
Cleared hypotheses: hj, hsi'
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx :
  x ∈ s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
j : ι
hj : ¬j = i
hsi' :
  Membership.mem (γ := Set X)
    (⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ) x
⊢ x ∈ (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
have hsi'₁ : ∀ (i_1 : ι) (j : ¬i_1 = i), x ∈ (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
Tactic: rw [Set.mem_iInter₂] at hsi'
Cleared hypotheses: hsi'
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
⊢ x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
have this : ∀ (j : ι) (hj : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
Tactic: have (j) (hj : j ≠ i) : x ∉ v j := by
      rw [hsi] at hx
      apply Set.notMem_of_mem_compl
      have hsi' : x ∈ (⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ) := Set.mem_of_mem_inter_right hx
      rw [ne_eq] at hj
      rw [Set.mem_iInter₂] at hsi'
      exact hsi' 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
this : ∀ (j : ι) (hj : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
⊢ x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
have obtain : ∀ (j : ι) (hj : x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j),
  x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
Tactic: obtain ⟨j, hj⟩ := Set.mem_iUnion.mp
      (v.subset_iUnion (Set.mem_of_mem_inter_left hx))
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
this : ∀ (j : ι) (hj : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
j : ι
hj : x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
⊢ j = i
have by_contra! : ∀ (h : j ≠ i), False
Tactic: by_contra! h
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
x : X
hx : x ∈ si
this : ∀ (j : ι) (hj : j ≠ i), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
j : ι
hj : x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
⊢ x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
have obtain : ∀ (hi : j ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))),
  let si :=
    s ∩ (⋃ (j_1 : ι), ⋃ (_ : j_1 ≠ j), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j_1)ᶜ;
  ∀
    (hsi :
      si = s ∩ ⋂ (i : ι), ⋂ (_ : ¬i = j), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i)ᶜ)
    (hsic : IsCompact si) (hx : x ∈ si)
    (this :
      ∀ (j_1 : ι) (hj : j_1 ≠ j), x ∉ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j_1),
    x ∈ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j
Tactic: obtain rfl : j = i := by
      by_contra! h
      exact this j h hj
Cleared hypotheses: i, hi, si, hsi, hsic, hx, this
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have this : si ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
Tactic: have : si ⊆ v i := by
    intro x hx
    have (j) (hj : j ≠ i) : x ∉ v j := by
      rw [hsi] at hx
      apply Set.notMem_of_mem_compl
      have hsi' : x ∈ (⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ) := Set.mem_of_mem_inter_right hx
      rw [ne_eq] at hj
      rw [Set.mem_iInter₂] at hsi'
      exact hsi' j hj
    obtain ⟨j, hj⟩ := Set.mem_iUnion.mp
      (v.subset_iUnion (Set.mem_of_mem_inter_left hx))
    obtain rfl : j = i := by
      by_contra! h
      exact this j h hj
    exact 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have obtain : ∀ (vi : Set X)
  (hvi :
    IsOpen vi ∧
      si ⊆ vi ∧
        closure vi ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i ∧
          IsCompact (closure vi)),
  ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
Tactic: obtain ⟨vi, hvi⟩ := exists_open_between_and_isCompact_closure hsic (v.isOpen i) this
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
vi : Set X
hvi :
  IsOpen vi ∧
    si ⊆ vi ∧
      closure vi ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i ∧ IsCompact (closure vi)
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have refine_1 : ∀ (i_1 : ι),
  IsOpen
    (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i vi
      i_1)
Tactic: refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ⟨?_, ?_⟩, ?_⟩
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
vi : Set X
hvi :
  IsOpen vi ∧
    si ⊆ vi ∧
      closure vi ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i ∧ IsCompact (closure vi)
refine_1 :
  ∀ (i_1 : ι),
    IsOpen
      (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i vi
        i_1)
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have refine_2 : s ⊆
  ⋃ (i_1 : ι),
    update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i vi
      i_1
Tactic: refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ⟨?_, ?_⟩, ?_⟩
-- 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 : T2Space X
inst_2 : LocallyCompactSpace X
v : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))
si : Set X :=
  s ∩ (⋃ (j : ι), ⋃ (_ : j ≠ i), v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) j)ᶜ
hsi :
  si =
    s ∩ ⋂ (i_1 : ι), ⋂ (_ : ¬i_1 = i), (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i
vi : Set X
hvi :
  IsOpen vi ∧
    si ⊆ vi ∧
      closure vi ⊆ v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i ∧ IsCompact (closure vi)
refine_1 :
  ∀ (i_1 : ι),
    IsOpen
      (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i vi
        i_1)
refine_2 :
  s ⊆
    ⋃ (i_1 : ι),
      update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i vi
        i_1
⊢ ∃ (v' : PartialRefinement u s fun (w : Set X) ↦ IsCompact (closure w)),
    v < v' ∧ IsCompact (closure (v'.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w)) i))
have refine_3 : ∀ {i_1 : ι},
  i_1 ∈ insert i (v.carrier (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) →
    closure
        (update (β := fun (a : ι) ↦ Set X) (v.toFun (u := u) (s := s) (p := fun (w : Set X) ↦ IsCompact (closure w))) i
          vi i_1) ⊆
      u i_1
Tactic: refine ⟨⟨update v i vi, insert i v.carrier, ?_, ?_, ?_, ?_, ?_⟩, ⟨?_, ?_⟩, ?_⟩