Encodable types

This file defines encodable (constructively countable) types as a typeclass. This is used to provide explicit encode/decode functions from and to ℕ, with the information that those functions are inverses of each other. The difference with Denumerable is that finite types are encodable. For infinite types, Encodable and Denumerable agree.

Main declarations

• Encodable α: States that there exists an explicit encoding function encode : α → ℕ with a partial inverse decode : ℕ → Option α.
• decode₂: Version of decode that is equal to none outside of the range of encode. Useful as we do not require this in the definition of decode.
• ULower α: Any encodable type has an equivalent type living in the lowest universe, namely a subtype of ℕ. ULower α finds it.

Implementation notes

The point of asking for an explicit partial inverse decode : ℕ → Option α to encode : α → ℕ is to make the range of encode decidable even when the finiteness of α is not.

class Encodable (α : Type u_1) : Type u_1

Constructively countable type. Made from an explicit injection encode : α → ℕ and a partial inverse decode : ℕ → Option α. Note that finite types are countable. See Denumerable if you wish to enforce infiniteness.

• encode : α → ℕ

  Encoding from Type α to ℕ

• decode : ℕ → Option α

  Decoding from ℕ to Option α

• encodek (a : α) : decode (encode a) = some a

  Invariant relationship between encoding and decoding

theorem Encodable.encode_injective {α : Type u_1} [Encodable α] : Function.Injective encode

@[simp]

theorem Encodable.encode_inj {α : Type u_1} [Encodable α] {a b : α} : encode a = encode b ↔ a = b

@[instance 400]

instance Encodable.countable {α : Type u_1} [Encodable α] : Countable α

theorem Encodable.surjective_decode_getD (α : Type u_3) [Encodable α] (d : α) : Function.Surjective fun (n : ℕ) => (decode n).getD d

@[deprecated Encodable.surjective_decode_getD (since := "2026-01-05")]

theorem Encodable.surjective_decode_iget (α : Type u_3) [Encodable α] [Inhabited α] : Function.Surjective fun (n : ℕ) => (decode n).getD default

def Encodable.decidableEqOfEncodable (α : Type u_3) [Encodable α] : DecidableEq α

An encodable type has decidable equality. Not set as an instance because this is usually not the best way to infer decidability.

def Encodable.ofLeftInjection {α : Type u_1} {β : Type u_2} [Encodable α] (f : β → α) (finv : α → Option β) (linv : ∀ (b : β), finv (f b) = some b) : Encodable β

If α is encodable and there is an injection f : β → α, then β is encodable as well.

def Encodable.ofLeftInverse {α : Type u_1} {β : Type u_2} [Encodable α] (f : β → α) (finv : α → β) (linv : ∀ (b : β), finv (f b) = b) : Encodable β

If α is encodable and f : β → α is invertible, then β is encodable as well.

def Encodable.ofEquiv {β : Type u_2} (α : Type u_3) [Encodable α] (e : β ≃ α) : Encodable β

Encodability is preserved by equivalence.

theorem Encodable.encode_ofEquiv {α : Type u_3} {β : Type u_4} [Encodable α] (e : β ≃ α) (b : β) : encode b = encode (e b)

theorem Encodable.decode_ofEquiv {α : Type u_3} {β : Type u_4} [Encodable α] (e : β ≃ α) (n : ℕ) : decode n = Option.map (⇑e.symm) (decode n)

instance Nat.encodable : Encodable ℕ

@[simp]

theorem Encodable.encode_nat (n : ℕ) : encode n = n

@[simp]

theorem Encodable.decode_nat (n : ℕ) : decode n = some n

@[instance 100]

instance IsEmpty.toEncodable {α : Type u_1} [IsEmpty α] : Encodable α

instance PUnit.encodable : Encodable PUnit.{u_3 + 1}

@[simp]

theorem Encodable.encode_star : encode PUnit.unit = 0

@[simp]

theorem Encodable.decode_unit_zero : decode 0 = some PUnit.unit

@[simp]

theorem Encodable.decode_unit_succ (n : ℕ) : decode n.succ = none

instance Option.encodable {α : Type u_3} [h : Encodable α] : Encodable (Option α)

If α is encodable, then so is Option α.

@[simp]

theorem Encodable.encode_none {α : Type u_1} [Encodable α] : encode none = 0

@[simp]

theorem Encodable.encode_some {α : Type u_1} [Encodable α] (a : α) : encode (some a) = (encode a).succ

@[simp]

theorem Encodable.decode_option_zero {α : Type u_1} [Encodable α] : decode 0 = some none

@[simp]

theorem Encodable.decode_option_succ {α : Type u_1} [Encodable α] (n : ℕ) : decode n.succ = Option.map some (decode n)

def Encodable.decode₂ (α : Type u_3) [Encodable α] (n : ℕ) : Option α

Failsafe variant of decode. decode₂ α n returns the preimage of n under encode if it exists, and returns none if it doesn't. This requirement could be imposed directly on decode but is not to help make the definition easier to use.

theorem Encodable.mem_decode₂' {α : Type u_1} [Encodable α] {n : ℕ} {a : α} : a ∈ decode₂ α n ↔ a ∈ decode n ∧ encode a = n

theorem Encodable.mem_decode₂ {α : Type u_1} [Encodable α] {n : ℕ} {a : α} : a ∈ decode₂ α n ↔ encode a = n

theorem Encodable.decode₂_eq_some {α : Type u_1} [Encodable α] {n : ℕ} {a : α} : decode₂ α n = some a ↔ encode a = n

@[simp]

theorem Encodable.decode₂_encode {α : Type u_1} [Encodable α] (a : α) : decode₂ α (encode a) = some a

theorem Encodable.decode₂_ne_none_iff {α : Type u_1} [Encodable α] {n : ℕ} : decode₂ α n ≠ none ↔ n ∈ Set.range encode

theorem Encodable.decode₂_is_partial_inv {α : Type u_1} [Encodable α] : Function.IsPartialInv encode (decode₂ α)

theorem Encodable.decode₂_inj {α : Type u_1} [Encodable α] {n : ℕ} {a₁ a₂ : α} (h₁ : a₁ ∈ decode₂ α n) (h₂ : a₂ ∈ decode₂ α n) : a₁ = a₂

theorem Encodable.encodek₂ {α : Type u_1} [Encodable α] (a : α) : decode₂ α (encode a) = some a

def Encodable.decidableRangeEncode (α : Type u_3) [Encodable α] : DecidablePred fun (x : ℕ) => x ∈ Set.range encode

The encoding function has decidable range.

def Encodable.equivRangeEncode (α : Type u_3) [Encodable α] : α ≃ ↑(Set.range encode)

An encodable type is equivalent to the range of its encoding function.

def Unique.encodable {α : Type u_1} [Unique α] : Encodable α

A type with unique element is encodable. This is not an instance to avoid diamonds.

def Encodable.encodeSum {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] : α ⊕ β → ℕ

Explicit encoding function for the sum of two encodable types.

def Encodable.decodeSum {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (n : ℕ) : Option (α ⊕ β)

Explicit decoding function for the sum of two encodable types.

instance Sum.encodable {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] : Encodable (α ⊕ β)

If α and β are encodable, then so is their sum.

@[simp]

theorem Encodable.encode_inl {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (a : α) : encode (Sum.inl a) = 2 * encode a

@[simp]

theorem Encodable.encode_inr {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (b : β) : encode (Sum.inr b) = 2 * encode b + 1

@[simp]

theorem Encodable.decode_sum_val {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (n : ℕ) : decode n = decodeSum n

instance Bool.encodable : Encodable Bool

@[simp]

theorem Encodable.encode_true : encode true = 1

@[simp]

theorem Encodable.encode_false : encode false = 0

@[simp]

theorem Encodable.decode_zero : decode 0 = some false

@[simp]

theorem Encodable.decode_one : decode 1 = some true

theorem Encodable.decode_ge_two (n : ℕ) (h : 2 ≤ n) : decode n = none

noncomputable instance Prop.encodable : Encodable Prop

def Encodable.encodeSigma {α : Type u_1} {γ : α → Type u_3} [Encodable α] [(a : α) → Encodable (γ a)] : Sigma γ → ℕ

Explicit encoding function for Sigma γ

def Encodable.decodeSigma {α : Type u_1} {γ : α → Type u_3} [Encodable α] [(a : α) → Encodable (γ a)] (n : ℕ) : Option (Sigma γ)

Explicit decoding function for Sigma γ

instance Sigma.encodable {α : Type u_1} {γ : α → Type u_3} [Encodable α] [(a : α) → Encodable (γ a)] : Encodable (Sigma γ)

@[simp]

theorem Encodable.decode_sigma_val {α : Type u_1} {γ : α → Type u_3} [Encodable α] [(a : α) → Encodable (γ a)] (n : ℕ) : decode n = (decode (Nat.unpair n).1).bind fun (a : α) => Option.map (Sigma.mk a) (decode (Nat.unpair n).2)

@[simp]

theorem Encodable.encode_sigma_val {α : Type u_1} {γ : α → Type u_3} [Encodable α] [(a : α) → Encodable (γ a)] (a : α) (b : γ a) : encode ⟨a, b⟩ = Nat.pair (encode a) (encode b)

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

If α and β are encodable, then so is their product.

@[simp]

theorem Encodable.decode_prod_val {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (n : ℕ) : decode n = (decode (Nat.unpair n).1).bind fun (a : α) => Option.map (Prod.mk a) (decode (Nat.unpair n).2)

@[simp]

theorem Encodable.encode_prod_val {α : Type u_1} {β : Type u_2} [Encodable α] [Encodable β] (a : α) (b : β) : encode (a, b) = Nat.pair (encode a) (encode b)

def Encodable.encodeSubtype {α : Type u_1} {P : α → Prop} [encA : Encodable α] : { a : α // P a } → ℕ

Explicit encoding function for a decidable subtype of an encodable type

def Encodable.decodeSubtype {α : Type u_1} {P : α → Prop} [encA : Encodable α] [decP : DecidablePred P] (v : ℕ) : Option { a : α // P a }

Explicit decoding function for a decidable subtype of an encodable type

instance Subtype.encodable {α : Type u_1} {P : α → Prop} [encA : Encodable α] [decP : DecidablePred P] : Encodable { a : α // P a }

A decidable subtype of an encodable type is encodable.

theorem Encodable.Subtype.encode_eq {α : Type u_1} {P : α → Prop} [encA : Encodable α] [decP : DecidablePred P] (a : Subtype P) : encode a = encode ↑a

instance Fin.encodable (n : ℕ) : Encodable (Fin n)

instance Int.encodable : Encodable ℤ

instance PNat.encodable : Encodable ℕ+

instance ULift.encodable {α : Type u_1} [Encodable α] : Encodable (ULift.{u_3, u_1} α)

The lift of an encodable type is encodable

instance PLift.encodable {α : Type u_1} [Encodable α] : Encodable (PLift α)

The lift of an encodable type is encodable.

noncomputable def Encodable.ofInj {α : Type u_1} {β : Type u_2} [Encodable β] (f : α → β) (hf : Function.Injective f) : Encodable α

If β is encodable and there is an injection f : α → β, then α is encodable as well.

noncomputable def Encodable.ofCountable (α : Type u_3) [Countable α] : Encodable α

If α is countable, then it has a (non-canonical) Encodable structure.

@[simp]

theorem Encodable.nonempty_encodable {α : Type u_1} : Nonempty (Encodable α) ↔ Countable α

theorem nonempty_encodable (α : Type u_1) [Countable α] : Nonempty (Encodable α)

See also nonempty_fintype, nonempty_denumerable.

instance instCountablePNat : Countable ℕ+

def ULower (α : Type u_1) [Encodable α] : Type

ULower α : Type is an equivalent type in the lowest universe, given Encodable α.

instance instDecidableEqULower {α : Type u_1} [Encodable α] : DecidableEq (ULower α)

instance instEncodableULower {α : Type u_1} [Encodable α] : Encodable (ULower α)

def ULower.equiv (α : Type u_1) [Encodable α] : α ≃ ULower α

The equivalence between the encodable type α and ULower α : Type.

def ULower.down {α : Type u_1} [Encodable α] (a : α) : ULower α

Lowers an a : α into ULower α.

instance ULower.instInhabited {α : Type u_1} [Encodable α] [Inhabited α] : Inhabited (ULower α)

def ULower.up {α : Type u_1} [Encodable α] (a : ULower α) : α

Lifts an a : ULower α into α.

@[simp]

theorem ULower.down_up {α : Type u_1} [Encodable α] {a : ULower α} : down a.up = a

@[simp]

theorem ULower.up_down {α : Type u_1} [Encodable α] {a : α} : (down a).up = a

@[simp]

theorem ULower.up_eq_up {α : Type u_1} [Encodable α] {a b : ULower α} : a.up = b.up ↔ a = b

@[simp]

theorem ULower.down_eq_down {α : Type u_1} [Encodable α] {a b : α} : down a = down b ↔ a = b

theorem ULower.ext {α : Type u_1} [Encodable α] {a b : ULower α} : a.up = b.up → a = b

theorem ULower.ext_iff {α : Type u_1} [Encodable α] {a b : ULower α} : a = b ↔ a.up = b.up

def Encodable.chooseX {α : Type u_1} {p : α → Prop} [Encodable α] [DecidablePred p] (h : ∃ (x : α), p x) : { a : α // p a }

Constructive choice function for a decidable subtype of an encodable type.

def Encodable.choose {α : Type u_1} {p : α → Prop} [Encodable α] [DecidablePred p] (h : ∃ (x : α), p x) : α

Constructive choice function for a decidable predicate over an encodable type.

theorem Encodable.choose_spec {α : Type u_1} {p : α → Prop} [Encodable α] [DecidablePred p] (h : ∃ (x : α), p x) : p (choose h)

theorem Encodable.axiom_of_choice {α : Type u_1} {β : α → Type u_2} {R : (x : α) → β x → Prop} [(a : α) → Encodable (β a)] [(x : α) → (y : β x) → Decidable (R x y)] (H : ∀ (x : α), ∃ (y : β x), R x y) : ∃ (f : (a : α) → β a), ∀ (x : α), R x (f x)

A constructive version of Classical.axiom_of_choice for Encodable types.

theorem Encodable.skolem {α : Type u_1} {β : α → Type u_2} {P : (x : α) → β x → Prop} [(a : α) → Encodable (β a)] [(x : α) → (y : β x) → Decidable (P x y)] : (∀ (x : α), ∃ (y : β x), P x y) ↔ ∃ (f : (a : α) → β a), ∀ (x : α), P x (f x)

A constructive version of Classical.skolem for Encodable types.

def Encodable.encode' (α : Type u_1) [Encodable α] : α ↪ ℕ

The encode function, viewed as an embedding.

instance Encodable.instAntisymmPreimageNatCoeEmbeddingEncode'Le {α : Type u_1} [Encodable α] : Std.Antisymm (⇑(encode' α) ⁻¹'o fun (x1 x2 : ℕ) => x1 ≤ x2)

instance Encodable.instTotalPreimageNatCoeEmbeddingEncode'Le {α : Type u_1} [Encodable α] : Std.Total (⇑(encode' α) ⁻¹'o fun (x1 x2 : ℕ) => x1 ≤ x2)

noncomputable def Directed.sequence {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] {r : β → β → Prop} (f : α → β) (hf : Directed r f) : ℕ → α

Given a Directed r function f : α → β defined on an encodable inhabited type, construct a noncomputable sequence such that r (f (x n)) (f (x (n + 1))) and r (f a) (f (x (encode a + 1)).

theorem Directed.sequence_mono_nat {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] {r : β → β → Prop} {f : α → β} (hf : Directed r f) (n : ℕ) : r (f (Directed.sequence f hf n)) (f (Directed.sequence f hf (n + 1)))

theorem Directed.rel_sequence {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] {r : β → β → Prop} {f : α → β} (hf : Directed r f) (a : α) : r (f a) (f (Directed.sequence f hf (Encodable.encode a + 1)))

theorem Directed.sequence_mono {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] [Preorder β] {f : α → β} (hf : Directed (fun (x1 x2 : β) => x1 ≤ x2) f) : Monotone (f ∘ Directed.sequence f hf)

theorem Directed.le_sequence {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] [Preorder β] {f : α → β} (hf : Directed (fun (x1 x2 : β) => x1 ≤ x2) f) (a : α) : f a ≤ f (Directed.sequence f hf (Encodable.encode a + 1))

theorem Directed.sequence_anti {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] [Preorder β] {f : α → β} (hf : Directed (fun (x1 x2 : β) => x1 ≥ x2) f) : Antitone (f ∘ Directed.sequence f hf)

theorem Directed.sequence_le {α : Type u_1} {β : Type u_2} [Encodable α] [Inhabited α] [Preorder β] {f : α → β} (hf : Directed (fun (x1 x2 : β) => x1 ≥ x2) f) (a : α) : f (Directed.sequence f hf (Encodable.encode a + 1)) ≤ f a

def Quotient.rep {α : Type u_1} {s : Setoid α} [DecidableRel fun (x1 x2 : α) => x1 ≈ x2] [Encodable α] (q : Quotient s) : α

Representative of an equivalence class. This is a computable version of Quot.out for a setoid on an encodable type.

theorem Quotient.rep_spec {α : Type u_1} {s : Setoid α} [DecidableRel fun (x1 x2 : α) => x1 ≈ x2] [Encodable α] (q : Quotient s) : ⟦q.rep⟧ = q

def encodableQuotient {α : Type u_1} {s : Setoid α} [DecidableRel fun (x1 x2 : α) => x1 ≈ x2] [Encodable α] : Encodable (Quotient s)

The quotient of an encodable space by a decidable equivalence relation is encodable.