Types that are empty

In this file we define a typeclass IsEmpty, which expresses that a type has no elements.

Main declaration

• IsEmpty: a typeclass that expresses that a type is empty.

class IsEmpty (α : Sort u) : Prop

IsEmpty α expresses that α is empty.

• false : ∀ (a : α), False

instance Empty.instIsEmpty : IsEmpty Empty

instance PEmpty.instIsEmpty : IsEmpty PEmpty.{u_1}

instance instIsEmptyFalse : IsEmpty False

instance Fin.isEmpty : IsEmpty (Fin 0)

instance Fin.isEmpty' : IsEmpty (Fin Nat.zero)

theorem Function.isEmpty {α : Sort u} {β : Sort v} [IsEmpty β] (f : α → β) : IsEmpty α

theorem Function.Surjective.isEmpty {α : Sort u} {β : Sort v} [IsEmpty α] {f : α → β} (hf : Surjective f) : IsEmpty β

instance instIsEmptyForallOfNonempty {α : Sort u} {p : α → Sort v} [∀ (x : α), IsEmpty (p x)] [h : Nonempty α] : IsEmpty ((x : α) → p x)

instance PProd.isEmpty_left {α : Sort u} {β : Sort v} [IsEmpty α] : IsEmpty (α ×' β)

instance PProd.isEmpty_right {α : Sort u} {β : Sort v} [IsEmpty β] : IsEmpty (α ×' β)

instance Prod.isEmpty_left {α : Type u_1} {β : Type u_2} [IsEmpty α] : IsEmpty (α × β)

instance Prod.isEmpty_right {α : Type u_1} {β : Type u_2} [IsEmpty β] : IsEmpty (α × β)

instance Quot.instIsEmpty {α : Sort u} [IsEmpty α] {r : α → α → Prop} : IsEmpty (Quot r)

instance Quotient.instIsEmpty {α : Sort u} [IsEmpty α] {s : Setoid α} : IsEmpty (Quotient s)

instance instIsEmptyPSum {α : Sort u} {β : Sort v} [IsEmpty α] [IsEmpty β] : IsEmpty (α ⊕' β)

instance instIsEmptySum {α : Type u_1} {β : Type u_2} [IsEmpty α] [IsEmpty β] : IsEmpty (α ⊕ β)

instance instIsEmptySubtype {α : Sort u} [IsEmpty α] (p : α → Prop) : IsEmpty (Subtype p)

subtypes of an empty type are empty

theorem Subtype.isEmpty_of_false {α : Sort u} {p : α → Prop} (hp : ∀ (a : α), ¬p a) : IsEmpty (Subtype p)

subtypes by an all-false predicate are false.

instance Subtype.isEmpty_false {α : Sort u} : IsEmpty { _a : α // False }

subtypes by false are false.

instance Sigma.isEmpty_left {α : Type u_1} [IsEmpty α] {E : α → Type v} : IsEmpty (Sigma E)

def isEmptyElim {α : Sort u} [IsEmpty α] {p : α → Sort v} (a : α) : p a

Eliminate out of a type that IsEmpty (without using projection notation).

theorem isEmpty_iff {α : Sort u} : IsEmpty α ↔ ∀ (a : α), False

def IsEmpty.elim {α : Sort u} : IsEmpty α → {p : α → Sort v} → (a : α) → p a

Eliminate out of a type that IsEmpty (using projection notation).

def IsEmpty.elim' {α : Sort u} {β : Sort v} (h : IsEmpty α) (a : α) : β

Non-dependent version of IsEmpty.elim. Helpful if the elaborator cannot elaborate h.elim a correctly.

theorem IsEmpty.prop_iff {p : Prop} : IsEmpty p ↔ ¬p

@[simp]

theorem IsEmpty.forall_iff {α : Sort u} [IsEmpty α] {p : α → Prop} : (∀ (a : α), p a) ↔ True

@[simp]

theorem IsEmpty.exists_iff {α : Sort u} [IsEmpty α] {p : α → Prop} : (∃ (a : α), p a) ↔ False

@[instance 100]

instance IsEmpty.instSubsingleton {α : Sort u} [IsEmpty α] : Subsingleton α