def Set (α : Type) : Type := α → Prop

instance : Membership α (Set α) :=
  ⟨fun s a => s a⟩

instance : HasSubset (Set α) :=
  ⟨fun s t => ∀ x, x ∈ s → x ∈ t⟩

instance : Union (Set α) :=
  ⟨fun s t => fun x => x ∈ s ∨ x ∈ t⟩

instance : Inter (Set α) :=
  ⟨fun s t => fun x => x ∈ s ∧ x ∈ t⟩

/-!
Exercícios da Aula 2: Lógica de Predicados e Conjuntos.
Substitua cada `sorry` por uma prova. As definições acima
vêm da aula.
-/
theorem exercise1 (α : Type) (P Q : α → Prop)
    (h : ∀ x, P x → Q x) (hP : ∀ x, P x) : ∀ x, Q x := by
  sorry

theorem exercise2 (α : Type) (P Q : α → Prop) :
    (∃ x, P x ∨ Q x) ↔ (∃ x, P x) ∨ (∃ x, Q x) := by
  sorry

theorem exercise3 (α : Type) (P : α → Prop) (Q : Prop)
    (h : ∃ x, P x → Q) (hP : ∀ x, P x) : Q := by
  sorry

theorem exercise4 (α : Type) (s t u : Set α)
    (hst : s ⊆ t) (htu : t ⊆ u) : s ⊆ u := by
  sorry

theorem exercise5 (α : Type) (s t u : Set α) :
    s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := by
  sorry