Countable sets

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def Set.Countable {α : Type u} (s : Set α) : Prop

A set is countable if there exists an encoding of the set into the natural numbers. An encoding is an injection with a partial inverse, which can be viewed as a constructive analogue of countability. (For the most part, theorems about Countable will be classical and Encodable will be constructive.)

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• Set.Countable s = Nonempty (Encodable ↑s)

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theorem Set.countable_coe_iff {α : Type u} {s : Set α} : Countable ↑s ↔ Set.Countable s

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theorem Set.to_countable {α : Type u} (s : Set α) [inst : Countable ↑s] : Set.Countable s

Prove Set.Countable from a Countable instance on the subtype.

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theorem Set.Countable.to_subtype {α : Type u} {s : Set α} : Set.Countable s → Countable ↑s

Restate Set.Countable as a Countable instance.

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theorem Countable.to_set {α : Type u} {s : Set α} : Countable ↑s → Set.Countable s

Restate Set.Countable as a Countable instance.

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theorem Set.countable_iff_exists_injective {α : Type u} {s : Set α} : Set.Countable s ↔ ∃ f, Function.Injective f

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theorem Set.countable_iff_exists_injOn {α : Type u} {s : Set α} : Set.Countable s ↔ ∃ f, Set.InjOn f s

A set s : Set α is countable if and only if there exists a function α → ℕ→ ℕ injective on s.

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def Set.Countable.toEncodable {α : Type u} {s : Set α} : Set.Countable s → Encodable ↑s

Convert Set.Countable s to Encodable s (noncomputable).

Equations

• Set.Countable.toEncodable = Classical.choice

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def Set.enumerateCountable {α : Type u} {s : Set α} (h : Set.Countable s) (default : α) : ℕ → α

Noncomputably enumerate elements in a set. The default value is used to extend the domain to all of ℕ.

Equations

• Set.enumerateCountable h default n = match Encodable.decode n with | some y => ↑y | none => default

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theorem Set.subset_range_enumerate {α : Type u} {s : Set α} (h : Set.Countable s) (default : α) : s ⊆ Set.range (Set.enumerateCountable h default)

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theorem Set.Countable.mono {α : Type u} {s₁ : Set α} {s₂ : Set α} (h : s₁ ⊆ s₂) : Set.Countable s₂ → Set.Countable s₁

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theorem Set.countable_range {β : Type v} {ι : Sort x} [inst : Countable ι] (f : ι → β) : Set.Countable (Set.range f)

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theorem Set.countable_iff_exists_subset_range {α : Type u} [inst : Nonempty α] {s : Set α} : Set.Countable s ↔ ∃ f, s ⊆ Set.range f

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theorem Set.countable_iff_exists_surjective {α : Type u} {s : Set α} (hs : Set.Nonempty s) : Set.Countable s ↔ ∃ f, Function.Surjective f

A non-empty set is countable iff there exists a surjection from the natural numbers onto the subtype induced by the set.

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theorem Set.Countable.exists_surjective {α : Type u} {s : Set α} (hs : Set.Nonempty s) : Set.Countable s → ∃ f, Function.Surjective f

Alias of the forward direction of Set.countable_iff_exists_surjective.

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theorem Set.countable_univ {α : Type u} [inst : Countable α] : Set.Countable Set.univ

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theorem Set.Countable.exists_eq_range {α : Type u} {s : Set α} (hc : Set.Countable s) (hs : Set.Nonempty s) : ∃ f, s = Set.range f

If s : Set α is a nonempty countable set, then there exists a map f : ℕ → α→ α such that s = range f.

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@[simp]

theorem Set.countable_empty {α : Type u} : Set.Countable ∅

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theorem Set.countable_singleton {α : Type u} (a : α) : Set.Countable {a}

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theorem Set.Countable.image {α : Type u} {β : Type v} {s : Set α} (hs : Set.Countable s) (f : α → β) : Set.Countable (f '' s)

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theorem Set.MapsTo.countable_of_injOn {α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : α → β} (hf : Set.MapsTo f s t) (hf' : Set.InjOn f s) (ht : Set.Countable t) : Set.Countable s

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theorem Set.Countable.preimage_of_injOn {α : Type u} {β : Type v} {s : Set β} (hs : Set.Countable s) {f : α → β} (hf : Set.InjOn f (f ⁻¹' s)) : Set.Countable (f ⁻¹' s)

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theorem Set.Countable.preimage {α : Type u} {β : Type v} {s : Set β} (hs : Set.Countable s) {f : α → β} (hf : Function.Injective f) : Set.Countable (f ⁻¹' s)

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theorem Set.exists_seq_supᵢ_eq_top_iff_countable {α : Type u} [inst : CompleteLattice α] {p : α → Prop} (h : ∃ x, p x) : (∃ s, ((n : ℕ) → p (s n)) ∧ (⨆ n, s n) = ⊤) ↔ ∃ S, Set.Countable S ∧ ((s : α) → s ∈ S → p s) ∧ supₛ S = ⊤

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theorem Set.exists_seq_cover_iff_countable {α : Type u} {p : Set α → Prop} (h : ∃ s, p s) : (∃ s, ((n : ℕ) → p (s n)) ∧ (Set.unionᵢ fun n => s n) = Set.univ) ↔ ∃ S, Set.Countable S ∧ ((s : Set α) → s ∈ S → p s) ∧ ⋃₀ S = Set.univ

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theorem Set.countable_of_injective_of_countable_image {α : Type u} {β : Type v} {s : Set α} {f : α → β} (hf : Set.InjOn f s) (hs : Set.Countable (f '' s)) : Set.Countable s

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theorem Set.countable_unionᵢ {α : Type u} {ι : Sort x} {t : ι → Set α} [inst : Countable ι] (ht : ∀ (i : ι), Set.Countable (t i)) : Set.Countable (Set.unionᵢ fun i => t i)

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theorem Set.countable_unionᵢ_iff {α : Type u} {ι : Sort x} [inst : Countable ι] {t : ι → Set α} : Set.Countable (Set.unionᵢ fun i => t i) ↔ ∀ (i : ι), Set.Countable (t i)

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theorem Set.Countable.bunionᵢ_iff {α : Type u} {β : Type v} {s : Set α} {t : (a : α) → a ∈ s → Set β} (hs : Set.Countable s) : Set.Countable (Set.unionᵢ fun a => Set.unionᵢ fun h => t a h) ↔ ∀ (a : α) (ha : a ∈ s), Set.Countable (t a ha)

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theorem Set.Countable.unionₛ_iff {α : Type u} {s : Set (Set α)} (hs : Set.Countable s) : Set.Countable (⋃₀ s) ↔ ∀ (a : Set α), a ∈ s → Set.Countable a

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theorem Set.Countable.bunionᵢ {α : Type u} {β : Type v} {s : Set α} {t : (a : α) → a ∈ s → Set β} (hs : Set.Countable s) : (∀ (a : α) (ha : a ∈ s), Set.Countable (t a ha)) → Set.Countable (Set.unionᵢ fun a => Set.unionᵢ fun h => t a h)

Alias of the reverse direction of Set.Countable.bunionᵢ_iff.

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theorem Set.Countable.unionₛ {α : Type u} {s : Set (Set α)} (hs : Set.Countable s) : (∀ (a : Set α), a ∈ s → Set.Countable a) → Set.Countable (⋃₀ s)

Alias of the reverse direction of Set.Countable.unionₛ_iff.

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theorem Set.countable_union {α : Type u} {s : Set α} {t : Set α} : Set.Countable (s ∪ t) ↔ Set.Countable s ∧ Set.Countable t

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theorem Set.Countable.union {α : Type u} {s : Set α} {t : Set α} (hs : Set.Countable s) (ht : Set.Countable t) : Set.Countable (s ∪ t)

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theorem Set.Countable.of_diff {α : Type u} {s : Set α} {t : Set α} (h : Set.Countable (s \ t)) (ht : Set.Countable t) : Set.Countable s

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theorem Set.countable_insert {α : Type u} {s : Set α} {a : α} : Set.Countable (insert a s) ↔ Set.Countable s

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theorem Set.Countable.insert {α : Type u} {s : Set α} (a : α) (h : Set.Countable s) : Set.Countable (insert a s)

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theorem Set.Finite.countable {α : Type u} {s : Set α} : Set.Finite s → Set.Countable s

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theorem Set.Countable.of_subsingleton {α : Type u} [inst : Subsingleton α] (s : Set α) : Set.Countable s

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theorem Set.Subsingleton.countable {α : Type u} {s : Set α} (hs : Set.Subsingleton s) : Set.Countable s

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theorem Set.countable_isTop (α : Type u_1) [inst : PartialOrder α] : Set.Countable { x | IsTop x }

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theorem Set.countable_isBot (α : Type u_1) [inst : PartialOrder α] : Set.Countable { x | IsBot x }

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theorem Set.countable_setOf_finite_subset {α : Type u} {s : Set α} (hs : Set.Countable s) : Set.Countable { t | Set.Finite t ∧ t ⊆ s }

The set of finite subsets of a countable set is countable.

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theorem Set.countable_univ_pi {α : Type u} {π : α → Type u_1} [inst : Finite α] {s : (a : α) → Set (π a)} (hs : ∀ (a : α), Set.Countable (s a)) : Set.Countable (Set.pi Set.univ s)

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theorem Set.countable_pi {α : Type u} {π : α → Type u_1} [inst : Finite α] {s : (a : α) → Set (π a)} (hs : ∀ (a : α), Set.Countable (s a)) : Set.Countable { f | ∀ (a : α), f a ∈ s a }

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theorem Set.Countable.prod {α : Type u} {β : Type v} {s : Set α} {t : Set β} (hs : Set.Countable s) (ht : Set.Countable t) : Set.Countable (s ×ˢ t)

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theorem Set.Countable.image2 {α : Type u} {β : Type v} {γ : Type w} {s : Set α} {t : Set β} (hs : Set.Countable s) (ht : Set.Countable t) (f : α → β → γ) : Set.Countable (Set.image2 f s t)

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theorem Finset.countable_toSet {α : Type u} (s : Finset α) : Set.Countable ↑s