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.

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class IsEmpty (α : Sort u_1) :

Prop

false : α → False

IsEmpty α expresses that α is empty.

Instances

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instance instIsEmptyEmpty :

IsEmpty Empty

Equations

instIsEmptyEmpty = instIsEmptyEmpty.proof_1

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instance instIsEmptyPEmpty :

IsEmpty PEmpty

Equations

instIsEmptyPEmpty = instIsEmptyPEmpty.proof_1

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instance instIsEmptyFalse :

IsEmpty False

Equations

instIsEmptyFalse = instIsEmptyFalse.proof_1

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instance instIsEmptyFinOfNatNatInstOfNatNat :

IsEmpty (Fin 0)

Equations

instIsEmptyFinOfNatNatInstOfNatNat = instIsEmptyFinOfNatNatInstOfNatNat.proof_1

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theorem Function.isEmpty {α : Sort u_2} {β : Sort u_1} [inst : IsEmpty β] (f : α → β) :

IsEmpty α

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instance instIsEmptyForAll {α : Sort u_1} {p : α → Sort u_2} [h : Nonempty α] [inst : ∀ (x : α), IsEmpty (p x)] :

IsEmpty ((x : α) → p x)

Equations

@instIsEmptyForAll = @instIsEmptyForAll.proof_1

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instance PProd.isEmpty_left {α : Sort u_1} {β : Sort u_2} [inst : IsEmpty α] :

IsEmpty (PProd α β)

Equations

@PProd.isEmpty_left = @PProd.isEmpty_left.proof_1

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instance PProd.isEmpty_right {α : Sort u_1} {β : Sort u_2} [inst : IsEmpty β] :

IsEmpty (PProd α β)

Equations

@PProd.isEmpty_right = @PProd.isEmpty_right.proof_1

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instance Prod.isEmpty_left {α : Type u_1} {β : Type u_2} [inst : IsEmpty α] :

IsEmpty (α × β)

Equations

@Prod.isEmpty_left = @Prod.isEmpty_left.proof_1

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instance Prod.isEmpty_right {α : Type u_1} {β : Type u_2} [inst : IsEmpty β] :

IsEmpty (α × β)

Equations

@Prod.isEmpty_right = @Prod.isEmpty_right.proof_1

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instance instIsEmptyPSum {α : Sort u_1} {β : Sort u_2} [inst : IsEmpty α] [inst : IsEmpty β] :

IsEmpty (α ⊕' β)

Equations

@instIsEmptyPSum = @instIsEmptyPSum.proof_1

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instance instIsEmptySum {α : Type u_1} {β : Type u_2} [inst : IsEmpty α] [inst : IsEmpty β] :

IsEmpty (α ⊕ β)

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@instIsEmptySum = @instIsEmptySum.proof_1

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instance instIsEmptySubtype {α : Sort u_1} [inst : IsEmpty α] (p : α → Prop) :

IsEmpty (Subtype p)

subtypes of an empty type are empty

Equations

@instIsEmptySubtype = @instIsEmptySubtype.proof_1

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theorem Subtype.isEmpty_of_false {α : Sort u_1} {p : α → Prop} (hp : ∀ (a : α), ¬p a) :

IsEmpty (Subtype p)

subtypes by an all-false predicate are false.

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instance Subtype.isEmpty_false {α : Sort u_1} :

IsEmpty { _a // False }

subtypes by false are false.

Equations

@Subtype.isEmpty_false = @Subtype.isEmpty_false.proof_1

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instance Sigma.isEmpty_left {α : Type u_1} [inst : IsEmpty α] {E : α → Type u_2} :

IsEmpty (Sigma E)

Equations

@Sigma.isEmpty_left = @Sigma.isEmpty_left.proof_1

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def isEmptyElim {α : Sort u_1} [inst : IsEmpty α] {p : α → Sort u_2} (a : α) :

p a

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

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isEmptyElim a = False.elim (_ : False)

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theorem isEmpty_iff {α : Sort u_1} :

IsEmpty α ↔ α → False

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def IsEmpty.elim {α : Sort u} :

IsEmpty α → {p : α → Sort u_1} → (a : α) → p a

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

Equations

IsEmpty.elim x a = isEmptyElim a

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def IsEmpty.elim' {α : Sort u_1} {β : Sort u_2} (h : IsEmpty α) (a : α) :

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

Equations

IsEmpty.elim' h a = False.elim (_ : False)

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theorem IsEmpty.prop_iff {p : Prop} :

IsEmpty p ↔ ¬p

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@[simp]

theorem IsEmpty.forall_iff {α : Sort u_1} [inst : IsEmpty α] {p : α → Prop} :

((a : α) → p a) ↔ True

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@[simp]

theorem IsEmpty.exists_iff {α : Sort u_1} [inst : IsEmpty α] {p : α → Prop} :

(∃ a, p a) ↔ False

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instance IsEmpty.instSubsingleton {α : Sort u_1} [inst : IsEmpty α] :

Subsingleton α

Equations

@IsEmpty.instSubsingleton = @IsEmpty.instSubsingleton.proof_1

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@[simp]

theorem not_nonempty_iff {α : Sort u_1} :

¬Nonempty α ↔ IsEmpty α

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@[simp]

theorem not_isEmpty_iff {α : Sort u_1} :

¬IsEmpty α ↔ Nonempty α

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@[simp]

theorem isEmpty_Prop {p : Prop} :

IsEmpty p ↔ ¬p

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@[simp]

theorem isEmpty_pi {α : Sort u_2} {π : α → Sort u_1} :

IsEmpty ((a : α) → π a) ↔ ∃ a, IsEmpty (π a)

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@[simp]

theorem isEmpty_sigma {α : Type u_1} {E : α → Type u_2} :

IsEmpty (Sigma E) ↔ ∀ (a : α), IsEmpty (E a)

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theorem isEmpty_psigma {α : Sort u_1} {E : α → Sort u_2} :

IsEmpty (PSigma E) ↔ ∀ (a : α), IsEmpty (E a)

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theorem isEmpty_subtype {α : Sort u_1} (p : α → Prop) :

IsEmpty (Subtype p) ↔ ∀ (x : α), ¬p x

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@[simp]

theorem isEmpty_prod {α : Type u_1} {β : Type u_2} :

IsEmpty (α × β) ↔ IsEmpty α ∨ IsEmpty β

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@[simp]

theorem isEmpty_pprod {α : Sort u_2} {β : Sort u_1} :

IsEmpty (PProd α β) ↔ IsEmpty α ∨ IsEmpty β

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@[simp]

theorem isEmpty_sum {α : Type u_1} {β : Type u_2} :

IsEmpty (α ⊕ β) ↔ IsEmpty α ∧ IsEmpty β

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@[simp]

theorem isEmpty_psum {α : Sort u_1} {β : Sort u_2} :

IsEmpty (α ⊕' β) ↔ IsEmpty α ∧ IsEmpty β

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@[simp]

theorem isEmpty_ulift {α : Type u_1} :

IsEmpty (ULift α) ↔ IsEmpty α

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@[simp]

theorem isEmpty_plift {α : Sort u_1} :

IsEmpty (PLift α) ↔ IsEmpty α

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theorem wellFounded_of_isEmpty {α : Sort u_1} [inst : IsEmpty α] (r : α → α → Prop) :

WellFounded r

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theorem isEmpty_or_nonempty (α : Sort u_1) :

IsEmpty α ∨ Nonempty α

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@[simp]

theorem not_isEmpty_of_nonempty (α : Sort u_1) [h : Nonempty α] :

¬IsEmpty α

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theorem Function.extend_of_isEmpty {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} [inst : IsEmpty α] (f : α → β) (g : α → γ) (h : β → γ) :

Function.extend f g h = h