awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (hypothesis)
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
⊢ False
have h₇ : 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
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (hypothesis)
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) = ∅
⊢ False
have h₄ : dissipate s (max j n) = dissipate s' (max j n)
Tactic: specialize h₃ (max j n) (Nat.le_max_right j n)
Cleared hypotheses: h₃
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (hypothesis)
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)
⊢ False
have hd₁ : (dissipate s (max j n)).Nonempty
Tactic: specialize hd (max j n)
Cleared hypotheses: hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₄ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₅ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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)),
  Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₆ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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)) (n : ℕ),
  (dissipate s' n).Nonempty
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₆ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)) (n : ℕ),
    (dissipate s' n).Nonempty
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₇ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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 = ∅), False
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₆ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)) (n : ℕ),
    (dissipate s' n).Nonempty
neg₇ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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 = ∅), False
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₈ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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), False
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₆ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)) (n : ℕ),
    (dissipate s' n).Nonempty
neg₇ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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 = ∅), False
neg₈ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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), False
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₉ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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) = ∅), False
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
neg neg₁ neg₂ neg₃ : Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₄ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let s' : (i : ℕ) → Set α := fun (i : ℕ) ↦ if s i ∈ S then s i else s n;
  ∀ (h₁ : ∀ (i : ℕ), s' i ∈ S), Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₅ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)),
    Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
neg₆ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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)) (n : ℕ),
    (dissipate s' n).Nonempty
neg₇ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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 = ∅), False
neg₈ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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), False
neg₉ :
  let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
  let 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) = ∅), False
⊢ Set.Nonempty.{u_1} (α := α) (⋂ (i : ℕ), s i)
have neg₁₀ : let n := Nat.find (p := fun (n : ℕ) ↦ s n ∈ S) h₀;
let 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)),
  False
Tactic: classical
  let n := Nat.find h₀
  let s' := fun i ↦ if s i ∈ S then s i else s n
  have h₁ : ∀ i, s' i ∈ S := by grind
  have h₂ : ⋂ i, s i = ⋂ i, s' i := by ext; simp; grind
  apply h₂ ▸ h s' h₁
  by_contra! ⟨j, hj⟩
  have h₃ (v : ℕ) (hv : n ≤ v) : dissipate s v = dissipate s' v := by ext; simp; grind
  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
  specialize h₃ (max j n) (Nat.le_max_right j n)
  specialize hd (max j n)
  simp [h₃, h₇] at hd
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
Goal:
α : Type u_1
S : Set (Set α)
⊢ 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)
have 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)
Tactic: simp_rw [IsCompactSystem.iff_nonempty_iInter]
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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)
have 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)
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 ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
⊢ ∀ (n : ℕ), Set.Nonempty.{u_1} (α := α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k)
have simp_rw : ∀ (h' : ∀ (n : ℕ), dissipate C n ≠ ∅) (n : ℕ), Ne.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅
Tactic: simp_rw [Set.nonempty_iff_ne_empty] at h' ⊢
Cleared hypotheses: h'
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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) ∅
have refine : ∀ (n : ℕ) (g : Eq.{u_1 + 1} (α := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) ∅), dissipate C n = ∅
Tactic: refine fun n g ↦ h' n ?_
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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) ∅
⊢ dissipate C n = ∅
have 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) ∅
Tactic: simp_rw [← subset_empty_iff, dissipate] at g ⊢
Cleared hypotheses: g
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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
⊢ Membership.mem (γ := Set α) (⋂ (y : ℕ), ⋂ (_ : y ≤ n), C y) x →
    Membership.mem (γ := Set α) (⋂ (k : ℕ), ⋂ (_ : k < n), C k) x
have simp : (∀ (i : ℕ) (i_1 : i ≤ n), x ∈ C i) → ∀ (i : ℕ) (i_1 : i < n), x ∈ C i
Tactic: simp
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
Goal:
α : Type u_1
S : Set (Set α)
hpi : IsPiSystem S
⊢ 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 = ∅
have 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 = ∅
Tactic: rw [← isCompactSystem_insert_empty_iff]
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
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 = ∅
have refine_1 : ∀ (h : IsCompactSystem (insert ∅ S)) (C : ℕ → Set α) (hdi : Directed (fun (x1 x2 : Set α) ↦ x1 ⊇ x2) C)
  (hi : ∀ (i : ℕ), C i ∈ S), Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅ → ∃ (n : ℕ), C n = ∅
Tactic: refine ⟨fun h ↦ fun C hdi hi ↦ ?_, fun h C h1 h2 ↦ ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Compactness.CompactSystem (goal)
Goal:
α : Type u_1
S : Set (Set α)
hpi : IsPiSystem S
refine_1 :
  ∀ (h : IsCompactSystem (insert ∅ S)) (C : ℕ → Set α) (hdi : Directed (fun (x1 x2 : Set α) ↦ x1 ⊇ x2) C)
    (hi : ∀ (i : ℕ), C i ∈ S), Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅ → ∃ (n : ℕ), C n = ∅
⊢ 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 = ∅
have refine_2 : ∀
  (h :
    ∀ (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 = ∅)
  (C : ℕ → Set α) (h1 : ∀ (i : ℕ), C i ∈ insert ∅ S) (h2 : Eq.{u_1 + 1} (α := Set α) (⋂ (i : ℕ), C i) ∅),
  ∃ (n : ℕ), dissipate C n = ∅
Tactic: refine ⟨fun h ↦ fun C hdi hi ↦ ?_, fun h C h1 h2 ↦ ?_⟩