awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h : IsCompactSystem S
inl : ∀ (hα : IsEmpty α), IsCompactSystem (insert Set.univ S)
inr : Nonempty α → IsCompactSystem (insert Set.univ S)
⊢ IsCompactSystem (insert Set.univ S)

Tactic:
rcases isEmpty_or_nonempty α with hα | _

Premises used: isEmpty_or_nonempty
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h : IsCompactSystem S
hα : IsEmpty α
⊢ IsCompactSystem (insert Set.univ S)

Tactic:
simp
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h : IsCompactSystem S
h_1 : Nonempty α
inr :
  ∀
    (h :
      ∀ (C : ℕ → Set α),
        (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)),
    Nonempty α →
      ∀ (C : ℕ → Set α),
        (∀ (i : ℕ), C i ∈ insert Set.univ S) →
          (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
⊢ IsCompactSystem (insert Set.univ S)

Tactic:
rw [IsCompactSystem.iff_nonempty_iInter] at h ⊢

Premises used: IsCompactSystem.iff_nonempty_iInter
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
pos : ∀ (h₀ : ∀ (n : ℕ), s n ∉ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg : ∀ (h₀ : ∃ (n : ℕ), s n ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)

Tactic:
by_cases! h₀ : ∀ n, s n ∉ S
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∀ (n : ℕ), s n ∉ S
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)

Tactic:
simp_all
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
⊢ ∀ (i : ℕ), s' i ∈ S

Tactic:
grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
⊢ ∀ (i : ℕ), s' i ∈ S

Tactic:
have h₁ : ∀ i, s' i ∈ S := by grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : ∀ (x : α), Membership.mem (γ := Set α) (⋂ (i : ℕ), s i) x ↔ Membership.mem (γ := Set α) (⋂ (i : ℕ), s' i) x
⊢ Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)

Tactic:
ext
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
x : α
h₂ : (∀ (i : ℕ), x ∈ s i) ↔ ∀ (i : ℕ), x ∈ s' i
⊢ Membership.mem (γ := Set α) (⋂ (i : ℕ), s i) x ↔ Membership.mem (γ := Set α) (⋂ (i : ℕ), s' i) x

Tactic:
simp
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
x : α
⊢ (∀ (i : ℕ), x ∈ s i) ↔ ∀ (i : ℕ), x ∈ s' i

Tactic:
grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
⊢ Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)

Tactic:
have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
neg : ∀ (n : ℕ), (dissipate s' n).Nonempty
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)

Tactic:
apply h₂ ▸ h s' h₁
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
neg : ∀ (j : ℕ) (hj : dissipate s' j = ∅), False
⊢ ∀ (n : ℕ), (dissipate s' n).Nonempty

Tactic:
by_contra! ⟨j, hj⟩
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
v : ℕ
hv : n ≤ v
h₃ : ∀ (x : α), x ∈ dissipate s v ↔ x ∈ dissipate s' v
⊢ dissipate s v = dissipate s' v

Tactic:
ext
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
v : ℕ
hv : n ≤ v
x : α
h₃ : (∀ y ≤ v, x ∈ s y) ↔ ∀ y ≤ v, x ∈ s' y
⊢ x ∈ dissipate s v ↔ x ∈ dissipate s' v

Tactic:
simp
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
v : ℕ
hv : n ≤ v
x : α
⊢ (∀ y ≤ v, x ∈ s y) ↔ ∀ y ≤ v, x ∈ s' y

Tactic:
grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
⊢ ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v

Tactic:
have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v
rw :
  ∀ (hj : dissipate s' j ⊆ ∅) (h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v), dissipate s' (max j n) ⊆ ∅
⊢ dissipate s' (max j n) = ∅

Tactic:
rw [← subset_empty_iff] at hj ⊢

Premises used: Set.subset_empty_iff
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j ⊆ ∅
h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v
⊢ dissipate s' (max j n) ⊆ ∅

Tactic:
(antitone_dissipate (Nat.le_max_left j n)).trans hj

Premises used: Nat.le_max_left, Set.antitone_dissipate, LE.le.trans
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j ⊆ ∅
h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v
⊢ dissipate s' (max j n) ⊆ ∅

Tactic:
exact (antitone_dissipate (Nat.le_max_left j n)).trans hj

Premises used: Nat.le_max_left, Set.antitone_dissipate, LE.le.trans
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v
⊢ dissipate s' (max j n) = ∅

Tactic:
have h₇ : dissipate s' (max j n) = ∅ := by
    rw [← subset_empty_iff] at hj ⊢
    exact (antitone_dissipate (Nat.le_max_left j n)).trans hj
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
h₃ : ∀ (v : ℕ) (hv : n ≤ v), dissipate s v = dissipate s' v
h₇ : dissipate s' (max j n) = ∅
⊢ dissipate s (max j n) = dissipate s' (max j n)

Tactic:
specialize h₃ (max j n) (Nat.le_max_right j n)

Premises used: Nat.le_max_right, Max.max
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
hd : ∀ (n : ℕ), (dissipate s n).Nonempty
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
h₇ : dissipate s' (max j n) = ∅
h₃ : dissipate s (max j n) = dissipate s' (max j n)
⊢ (dissipate s (max j n)).Nonempty

Tactic:
specialize hd (max j n)

Premises used: Max.max
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
h_1 : Nonempty α
s : ℕ → Set α
h' : ∀ (i : ℕ), s i ∈ insert Set.univ S
h₀ : ∃ (n : ℕ), s n ∈ S
n : ℕ := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀
s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n
h₁ : ∀ (i : ℕ), s' i ∈ S
h₂ : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), s i) (⋂ (i : ℕ), s' i)
j : ℕ
hj : dissipate s' j = ∅
h₇ : dissipate s' (max j n) = ∅
h₃ : dissipate s (max j n) = dissipate s' (max j n)
hd : (dissipate s (max j n)).Nonempty
⊢ False

Tactic:
simp [h₃, h₇] at hd
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
simp_rw :
  (∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)) ↔
    ∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) →
        (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
          Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
⊢ IsCompactSystem S ↔
    ∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) →
        (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
          Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)

Tactic:
simp_rw [IsCompactSystem.iff_nonempty_iInter]

Premises used: IsCompactSystem.iff_nonempty_iInter
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
⊢ (∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)) ↔
    ∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) →
        (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
          Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)

Tactic:
⟨fun h C hi h'↦ h C hi (fun n ↦ dissipate_eq_biInter_lt ▸ (h' (n + 1))),
    fun h C hi h' ↦ h C hi ?_⟩

Premises used: Set.dissipate_eq_biInter_lt, Eq.rec, HAdd.hAdd, Iff.intro, OfNat.ofNat, Eq.symm
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
refine :
  ∀
    (h :
      ∀ (C : ℕ → Set α),
        (∀ (i : ℕ), C i ∈ S) →
          (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
            Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i))
    (C : ℕ → Set α) (hi : ∀ (i : ℕ), C i ∈ S) (h' : ∀ (n : ℕ), (dissipate C n).Nonempty) (n : ℕ),
    Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)
⊢ (∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)) ↔
    ∀ (C : ℕ → Set α),
      (∀ (i : ℕ), C i ∈ S) →
        (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
          Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)

Tactic:
refine ⟨fun h C hi h'↦ h C hi (fun n ↦ dissipate_eq_biInter_lt ▸ (h' (n + 1))),
    fun h C hi h' ↦ h C hi ?_⟩
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), (dissipate C n).Nonempty
simp_rw : ∀ (h' : ∀ (n : ℕ), dissipate C n ≠ ∅) (n : ℕ), Ne.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
⊢ ∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)

Tactic:
simp_rw [Set.nonempty_iff_ne_empty] at h' ⊢

Premises used: Set.nonempty_iff_ne_empty
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
⊢ ∀ (n : ℕ), Ne.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅

Tactic:
fun n g ↦ h' n ?_

Premises used: Not
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
refine : ∀ (n : ℕ) (g : Eq.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅), dissipate C n = ∅
⊢ ∀ (n : ℕ), Ne.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅

Tactic:
refine fun n g ↦ h' n ?_
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
n : ℕ
g : Eq.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
simp_rw :
  ∀ (g : Subset.{u_1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅),
    Subset.{u_1} (α := Set α) (⋂ (y : ℕ), ⋂ (_ : y ≤ n), C y) ∅
⊢ dissipate C n = ∅

Tactic:
simp_rw [← subset_empty_iff, dissipate] at g ⊢

Premises used: Set.dissipate, Set.subset_empty_iff
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
n : ℕ
g : Subset.{u_1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
x : α
simp : (∀ (i : ℕ) (i_1 : i ≤ n), x ∈ C i) → ∀ (i : ℕ) (i_1 : i < n), x ∈ C i
⊢ Membership.mem (γ := Set α) (⋂ (y : ℕ), ⋂ (_ : y ≤ n), C y) x →
    Membership.mem (γ := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) x

Tactic:
simp
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
n : ℕ
g : Subset.{u_1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
x : α
⊢ (∀ (i : ℕ) (i_1 : i ≤ n), x ∈ C i) → ∀ (i : ℕ) (i_1 : i < n), x ∈ C i

Tactic:
grind
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
h :
  ∀ (C : ℕ → Set α),
    (∀ (i : ℕ), C i ∈ S) →
      (∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)) →
        Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), C i)
C : ℕ → Set α
hi : ∀ (i : ℕ), C i ∈ S
h' : ∀ (n : ℕ), dissipate C n ≠ ∅
n : ℕ
g : Subset.{u_1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
⊢ Subset.{u_1} (α := Set α) (⋂ (y : ℕ), ⋂ (_ : y ≤ n), C y) ∅

Tactic:
exact le_trans (fun x ↦ by simp; grind) g

Premises used: le_trans
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
⊢ IsCompactSystem (insert ∅ S) ↔ IsCompactSystem S

Tactic:
⟨fun h ↦ h.mono (subset_insert _ _), .insert_empty⟩

Premises used: IsCompactSystem.insert_empty, IsCompactSystem.mono, Iff.intro, EmptyCollection.emptyCollection, Set.subset_insert
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
⊢ IsCompactSystem (insert Set.univ S) ↔ IsCompactSystem S

Tactic:
⟨fun h ↦ h.mono (subset_insert _ _), .insert_univ⟩

Premises used: Set.univ, IsCompactSystem.mono, Iff.intro, Set.subset_insert, IsCompactSystem.insert_univ
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
hpi : IsPiSystem S
rw :
  IsCompactSystem (insert ∅ S) ↔
    ∀ (C : ℕ → Set α),
      Directed (fun (x1 x2 : Set α) ↦ x1 ⊇ x2) C →
        (∀ (i : ℕ), C i ∈ S) → Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅ → ∃ (n : ℕ), C n = ∅
⊢ IsCompactSystem S ↔
    ∀ (C : ℕ → Set α),
      Directed (fun (x1 x2 : Set α) ↦ x1 ⊇ x2) C →
        (∀ (i : ℕ), C i ∈ S) → Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅ → ∃ (n : ℕ), C n = ∅

Tactic:
rw [← isCompactSystem_insert_empty_iff]

Premises used: isCompactSystem_insert_empty_iff
-- Module: Mathlib.Topology.Compactness.CompactSystem
Goal:
α : Type u_1
S : Set (Set α)
hpi : IsPiSystem S
⊢ IsCompactSystem (insert ∅ S) ↔
    ∀ (C : ℕ → Set α),
      Directed (fun (x1 x2 : Set α) ↦ x1 ⊇ x2) C →
        (∀ (i : ℕ), C i ∈ S) → Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅ → ∃ (n : ℕ), C n = ∅

Tactic:
⟨fun h ↦ fun C hdi hi ↦ ?_, fun h C h1 h2 ↦ ?_⟩

Premises used: Iff.intro