Documentation

Mathlib.Logic.IsEmpty

Types that are empty #

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

Main declaration #

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class IsEmpty (α : Sort u_4) :
Prop

IsEmpty α expresses that α is empty.

Instances
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theorem IsEmpty.false {α : Sort u_4} [self : IsEmpty α] :
α → False
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instance instIsEmptyEmpty :
IsEmpty Empty
Equations
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instance instIsEmptyPEmpty :
IsEmpty PEmpty.{u_4}
Equations
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instance instIsEmptyFalse :
IsEmpty False
Equations
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instance Fin.isEmpty :
IsEmpty (Fin 0)
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instance Fin.isEmpty' :
IsEmpty (Fin Nat.zero)
Equations
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theorem Function.isEmpty {α : Sort u_1} {β : Sort u_2} [IsEmpty β] (f : α → β) :
IsEmpty α
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theorem Function.Surjective.isEmpty {α : Sort u_1} {β : Sort u_2} [IsEmpty α] {f : α → β} (hf : Function.Surjective f) :
IsEmpty β
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instance instIsEmptyForallOfNonempty {α : Sort u_1} {p : α → Sort u_4} [h : Nonempty α] [∀ (x : α), IsEmpty (p x)] :
IsEmpty ((x : α) → p x)
Equations
  • ⋯ = ⋯
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instance PProd.isEmpty_left {α : Sort u_1} {β : Sort u_2} [IsEmpty α] :
IsEmpty (PProd α β)
Equations
  • ⋯ = ⋯
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instance PProd.isEmpty_right {α : Sort u_1} {β : Sort u_2} [IsEmpty β] :
IsEmpty (PProd α β)
Equations
  • ⋯ = ⋯
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instance Prod.isEmpty_left {α : Type u_4} {β : Type u_5} [IsEmpty α] :
IsEmpty (α × β)
Equations
  • ⋯ = ⋯
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instance Prod.isEmpty_right {α : Type u_4} {β : Type u_5} [IsEmpty β] :
IsEmpty (α × β)
Equations
  • ⋯ = ⋯
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instance Quot.instIsEmpty {α : Sort u_4} [IsEmpty α] {r : α → α → Prop} :
IsEmpty (Quot r)
Equations
  • ⋯ = ⋯
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instance Quotient.instIsEmpty {α : Sort u_4} [IsEmpty α] {s : Setoid α} :
IsEmpty (Quotient s)
Equations
  • ⋯ = ⋯
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instance instIsEmptyPSum {α : Sort u_1} {β : Sort u_2} [IsEmpty α] [IsEmpty β] :
IsEmpty (α ⊕' β)
Equations
  • ⋯ = ⋯
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instance instIsEmptySum {α : Type u_4} {β : Type u_5} [IsEmpty α] [IsEmpty β] :
IsEmpty (α ⊕ β)
Equations
  • ⋯ = ⋯
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instance instIsEmptySubtype {α : Sort u_1} [IsEmpty α] (p : α → Prop) :
IsEmpty (Subtype p)

subtypes of an empty type are empty

Equations
  • ⋯ = ⋯
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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 : α // False }

subtypes by false are false.

Equations
  • ⋯ = ⋯
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instance Sigma.isEmpty_left {α : Type u_5} [IsEmpty α] {E : α → Type u_4} :
IsEmpty (Sigma E)
Equations
  • ⋯ = ⋯
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def isEmptyElim {α : Sort u_1} [IsEmpty α] {p : α → Sort u_4} (a : α) :
p a

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

Equations
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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_4} → (a : α) → p a

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

Equations
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def IsEmpty.elim' {α : Sort u_1} {β : Sort u_4} (h : IsEmpty α) (a : α) :
β

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

Equations
  • h.elim' a = ⋯.elim
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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} [IsEmpty α] {p : α → Prop} :
(∀ (a : α), p a) ↔ True
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@[simp]
theorem IsEmpty.exists_iff {α : Sort u_1} [IsEmpty α] {p : α → Prop} :
(∃ (a : α), p a) ↔ False
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@[instance 100]
instance IsEmpty.instSubsingleton {α : Sort u_1} [IsEmpty α] :
Subsingleton α
Equations
  • ⋯ = ⋯
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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_1} {π : α → Sort u_4} :
IsEmpty ((a : α) → π a) ↔ ∃ (a : α), IsEmpty (π a)
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theorem isEmpty_fun {α : Sort u_1} {β : Sort u_2} :
IsEmpty (α → β) ↔ Nonempty α ∧ IsEmpty β
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@[simp]
theorem nonempty_fun {α : Sort u_1} {β : Sort u_2} :
Nonempty (α → β) ↔ IsEmpty α ∨ Nonempty β
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@[simp]
theorem isEmpty_sigma {α : Type u_5} {E : α → Type u_4} :
IsEmpty (Sigma E) ↔ ∀ (a : α), IsEmpty (E a)
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@[simp]
theorem isEmpty_psigma {α : Sort u_5} {E : α → Sort u_4} :
IsEmpty (PSigma E) ↔ ∀ (a : α), IsEmpty (E a)
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@[simp]
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_4} {β : Type u_5} :
IsEmpty (α × β) ↔ IsEmpty α ∨ IsEmpty β
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@[simp]
theorem isEmpty_pprod {α : Sort u_1} {β : Sort u_2} :
IsEmpty (PProd α β) ↔ IsEmpty α ∨ IsEmpty β
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@[simp]
theorem isEmpty_sum {α : Type u_4} {β : Type u_5} :
IsEmpty (α ⊕ β) ↔ IsEmpty α ∧ IsEmpty β
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@[simp]
theorem isEmpty_psum {α : Sort u_4} {β : Sort u_5} :
IsEmpty (α ⊕' β) ↔ IsEmpty α ∧ IsEmpty β
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@[simp]
theorem isEmpty_ulift {α : Type u_4} :
IsEmpty (ULift.{u_5, u_4} α) ↔ IsEmpty α
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@[simp]
theorem isEmpty_plift {α : Sort u_4} :
IsEmpty (PLift α) ↔ IsEmpty α
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theorem wellFounded_of_isEmpty {α : Sort u_4} [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} [IsEmpty α] (f : α → β) (g : α → γ) (h : β → γ) :
Function.extend f g h = h