Definition of the Finite typeclass

This file defines a typeclass Finite saying that α : Sort* is finite. A type is Finite if it is equivalent to Fin n for some n. We also define Infinite α as a typeclass equivalent to ¬Finite α.

The Finite predicate has no computational relevance and, being Prop-valued, gets to enjoy proof irrelevance -- it represents the mere fact that the type is finite. While the Finite class also represents finiteness of a type, a key difference is that a Fintype instance represents finiteness in a computable way: it gives a concrete algorithm to produce a Finset whose elements enumerate the terms of the given type. As such, one generally relies on congruence lemmas when rewriting expressions involving Fintype instances.

Every Fintype instance automatically gives a Finite instance, see Fintype.finite, but not vice versa. Every Fintype instance should be computable since they are meant for computation. If it's not possible to write a computable Fintype instance, one should prefer writing a Finite instance instead.

Main definitions

• Finite α denotes that α is a finite type.
• Infinite α denotes that α is an infinite type.

Implementation notes

The definition of Finite α is not just Nonempty (Fintype α) since Fintype requires that α : Type*, and the definition in this module allows for α : Sort*. This means we can write the instance Finite.prop.

Tags

finite, fintype

class inductive Finite (α : Sort u_3) : Prop

A type is Finite if it is in bijective correspondence to some Fin n.

This is similar to Fintype, but Finite is a proposition rather than data. A particular benefit to this is that Finite instances are definitionally equal to one another (due to proof irrelevance) rather than being merely propositionally equal, and, furthermore, Finite instances generally avoid the need for Decidable instances. One other notable difference is that Finite allows there to be Finite p instances for all p : Prop, which is not allowed by Fintype due to universe constraints. An application of this is that Finite (x ∈ s → β x) follows from the general instance for pi types, assuming [∀ x, Finite (β x)]. Implementation note: this is a reason Finite α is not defined as Nonempty (Fintype α).

Every Fintype instance provides a Finite instance via Finite.of_fintype. Conversely, one can noncomputably create a Fintype instance from a Finite instance via Fintype.ofFinite. In a proof one might write

  have := Fintype.ofFinite α

to obtain such an instance.

Do not write noncomputable Fintype instances; instead write Finite instances and use this Fintype.ofFinite interface. The Fintype instances should be relied upon to be computable for evaluation purposes.

Theorems should use Finite instead of Fintype, unless definitions in the theorem statement require Fintype. Definitions should prefer Finite as well, unless it is important that the definitions are meant to be computable in the reduction or #eval sense.

• intro: ∀ {α : Sort u_3} {n : ℕ}, α ≃ Fin n → Finite α

theorem finite_iff_exists_equiv_fin {α : Sort u_3} : Finite α ↔ ∃ (n : ℕ), Nonempty (α ≃ Fin n)

theorem Finite.exists_equiv_fin (α : Sort u_3) [h : Finite α] : ∃ (n : ℕ), Nonempty (α ≃ Fin n)

theorem Finite.of_equiv {β : Sort u_2} (α : Sort u_3) [h : Finite α] (f : α ≃ β) : Finite β

theorem Equiv.finite_iff {α : Sort u_1} {β : Sort u_2} (f : α ≃ β) : Finite α ↔ Finite β

theorem Function.Bijective.finite_iff {α : Sort u_1} {β : Sort u_2} {f : α → β} (h : Function.Bijective f) : Finite α ↔ Finite β

theorem Finite.ofBijective {α : Sort u_1} {β : Sort u_2} [Finite α] {f : α → β} (h : Function.Bijective f) : Finite β

instance instFinitePLift {α : Sort u_1} [Finite α] : Finite (PLift α)

Equations

• ⋯ = ⋯

instance instFiniteULift {α : Type v} [Finite α] : Finite (ULift.{u, v} α)

Equations

• ⋯ = ⋯

class Infinite (α : Sort u_3) : Prop

A type is said to be infinite if it is not finite. Note that Infinite α is equivalent to IsEmpty (Fintype α) or IsEmpty (Finite α).

• not_finite : ¬Finite α

  assertion that α is ¬Finite

theorem Infinite.not_finite {α : Sort u_3} [self : Infinite α] : ¬Finite α

assertion that α is ¬Finite

@[simp]

theorem not_finite_iff_infinite {α : Sort u_1} : ¬Finite α ↔ Infinite α

@[simp]

theorem not_infinite_iff_finite {α : Sort u_1} : ¬Infinite α ↔ Finite α

theorem Equiv.infinite_iff {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) : Infinite α ↔ Infinite β

instance instInfinitePLift {α : Sort u_1} [Infinite α] : Infinite (PLift α)

Equations

• ⋯ = ⋯

instance instInfiniteULift {α : Type v} [Infinite α] : Infinite (ULift.{u, v} α)

Equations

• ⋯ = ⋯

theorem finite_or_infinite (α : Sort u_3) : Finite α ∨ Infinite α

theorem not_finite (α : Sort u_3) [Infinite α] [Finite α] : False

Infinite α is not Finite

theorem Finite.false {α : Sort u_1} [Infinite α] : Finite α → False

theorem Infinite.false {α : Sort u_1} [Finite α] : Infinite α → False

theorem Finite.of_not_infinite {α : Sort u_1} : ¬Infinite α → Finite α

Alias of the forward direction of not_infinite_iff_finite.

theorem Finite.not_infinite {α : Sort u_1} : Finite α → ¬Infinite α

Alias of the reverse direction of not_infinite_iff_finite.