Encodings

This file contains the definition of an encoding, a map from a type to strings in an alphabet, used in defining computability by Turing machines. It also contains several examples:

Examples

• encodingNatBool : a binary encoding of ℕ in a simple alphabet.
• encodingNatΓ' : a binary encoding of ℕ in the alphabet used for TM's.
• unaryEncodingNat : a unary encoding of ℕ
• encodingBoolBool : an encoding of Bool.
• encodingList : an encoding of List α in the alphabet α.
• encodingProd : an encoding of α × β from encodings of α and β.

structure Computability.Encoding (α : Type u) (Γ : Type v) : Type (max u v)

An encoding of a type in a certain alphabet, together with a decoding.

• encode : α → List Γ

  The encoding function

• decode : List Γ → Option α

  The decoding function

• decode_encode (x : α) : self.decode (self.encode x) = some x

  Decoding and encoding are inverses of each other.

theorem Computability.Encoding.encode_injective {α : Type u_1} {Γ : Type u_2} (e : Encoding α Γ) : Function.Injective e.encode

inductive Computability.Γ' : Type

A standard Turing machine alphabet, consisting of blank,bit0,bit1,bra,ket,comma.

• blank : Γ'
• bit (b : Bool) : Γ'
• bra : Γ'
• ket : Γ'
• comma : Γ'

def Computability.instDecidableEqΓ'.decEq (x✝ x✝¹ : Γ') : Decidable (x✝ = x✝¹)

@[instance_reducible]

instance Computability.instDecidableEqΓ' : DecidableEq Γ'

@[instance_reducible]

instance Computability.instFintypeΓ' : Fintype Γ'

@[instance_reducible]

instance Computability.inhabitedΓ' : Inhabited Γ'

def Computability.inclusionBoolΓ' : Bool → Γ'

The natural inclusion of Bool in Γ'.

def Computability.sectionΓ'Bool : Γ' → Bool

An arbitrary section of the natural inclusion of Bool in Γ'.

@[simp]

theorem Computability.sectionΓ'Bool_inclusionBoolΓ' {b : Bool} : sectionΓ'Bool (inclusionBoolΓ' b) = b

theorem Computability.inclusionBoolΓ'_injective : Function.Injective inclusionBoolΓ'

def Computability.encodePosNum : PosNum → List Bool

An encoding function of the positive binary numbers in Bool.

def Computability.encodeNum : Num → List Bool

An encoding function of the binary numbers in Bool.

def Computability.encodeNat (n : ℕ) : List Bool

An encoding function of ℕ in Bool.

def Computability.decodePosNum : List Bool → PosNum

A decoding function from List Bool to the positive binary numbers.

def Computability.decodeNum : List Bool → Num

A decoding function from List Bool to the binary numbers.

def Computability.decodeNat : List Bool → ℕ

A decoding function from List Bool to ℕ.

theorem Computability.encodePosNum_nonempty (n : PosNum) : encodePosNum n ≠ []

@[simp]

theorem Computability.decode_encodePosNum (n : PosNum) : decodePosNum (encodePosNum n) = n

@[simp]

theorem Computability.decode_encodeNum (n : Num) : decodeNum (encodeNum n) = n

@[simp]

theorem Computability.decode_encodeNat (n : ℕ) : decodeNat (encodeNat n) = n

def Computability.encodingNatBool : Encoding ℕ Bool

A binary Encoding of ℕ in Bool.

def Computability.encodingNatΓ' : Encoding ℕ Γ'

A binary Encoding of ℕ in Γ'.

def Computability.unaryEncodeNat : ℕ → List Bool

A unary encoding function of ℕ in Bool.

def Computability.unaryDecodeNat : List Bool → ℕ

A unary decoding function from List Bool to ℕ.

@[simp]

theorem Computability.unary_decode_encode_nat (n : ℕ) : unaryDecodeNat (unaryEncodeNat n) = n

def Computability.unaryEncodingNat : Encoding ℕ Bool

A unary Encoding of ℕ in Bool.

def Computability.encodeBool : Bool → List Bool

An encoding function of Bool in Bool.

def Computability.decodeBool : List Bool → Bool

A decoding function from List Bool to Bool.

@[simp]

theorem Computability.decode_encodeBool (b : Bool) : decodeBool (encodeBool b) = b

def Computability.encodingBoolBool : Encoding Bool Bool

An Encoding of Bool in Bool.

@[instance_reducible]

instance Computability.inhabitedEncoding : Inhabited (Encoding Bool Bool)

theorem Computability.Encoding.card_le_card_list {α : Type u} {Γ : Type v} (e : Encoding α Γ) : Cardinal.lift.{v, u} (Cardinal.mk α) ≤ Cardinal.lift.{u, v} (Cardinal.mk (List Γ))

theorem Computability.Encoding.card_le_aleph0 {α : Type u_1} {Γ : Type u_2} (e : Encoding α Γ) [Countable Γ] : Cardinal.mk α ≤ Cardinal.aleph0

def Computability.encodingList (α : Type) : Encoding (List α) α

An Encoding of a List α in alphabet α, encoded directly.

def Computability.encodingProd {α : Type u_1} {β : Type u_2} {Γ₁ : Type u_3} {Γ₂ : Type u_4} (ea : Encoding α Γ₁) (eb : Encoding β Γ₂) : Encoding (α × β) (Γ₁ ⊕ Γ₂)

Given an Encoding of α and β, constructs an Encoding of α × β by concatenating the encodings, mapping the symbols from the first encoding with Sum.inl and those from the second with Sum.inr.

Deprecated aliases for FinEncoding and unbundled Γ

@[reducible, deprecated "Use `Encoding α Γ` along with `[Fintype Γ]` instead" (since := "2026-05-07")]

def Computability.FinEncoding (α : Type u) {Γ : Type v} [Fintype Γ] : Type (max u v)

Deprecated: Use Encoding α Γ along with [Fintype Γ] instead.

@[reducible, deprecated "Γ is now an explicit parameter of `Encoding`" (since := "2026-05-07")]

def Computability.Encoding.Γ {α : Type u} {Γ : Type v} : Encoding α Γ → Type v

Deprecated: Γ is now an explicit parameter of Encoding.

@[reducible, deprecated "Use `inferInstanceAs (Fintype Γ)` instead" (since := "2026-05-07")]

def Computability.FinEncoding.ΓFin {α : Type u} {Γ : Type v} [h : Fintype Γ] : Encoding α Γ → Fintype Γ

Deprecated: Use inferInstanceAs (Fintype Γ) instead.

@[reducible, deprecated "Use the encoding directly" (since := "2026-05-07")]

def Computability.FinEncoding.toEncoding {α : Type u} {Γ : Type v} [Fintype Γ] (e : Encoding α Γ) : Encoding α Γ

Deprecated: Use the encoding directly.

@[reducible, inline, deprecated Computability.encodingNatBool (since := "2026-05-07")]

abbrev Computability.finEncodingNatBool : Encoding ℕ Bool

Deprecated alias for encodingNatBool.

@[reducible, inline, deprecated Computability.encodingNatΓ' (since := "2026-05-07")]

abbrev Computability.finEncodingNatΓ' : Encoding ℕ Γ'

Deprecated alias for encodingNatΓ'.

@[reducible, inline, deprecated Computability.unaryEncodingNat (since := "2026-05-07")]

abbrev Computability.unaryFinEncodingNat : Encoding ℕ Bool

Deprecated alias for unaryEncodingNat.

@[reducible, inline, deprecated Computability.encodingBoolBool (since := "2026-05-07")]

abbrev Computability.finEncodingBoolBool : Encoding Bool Bool

Deprecated alias for encodingBoolBool.

@[reducible, deprecated Computability.encodingList (since := "2026-05-07")]

def Computability.finEncodingList (α : Type) [Fintype α] : Encoding (List α) α

Deprecated alias for encodingList.

@[reducible, deprecated Computability.encodingProd (since := "2026-05-07")]

def Computability.finEncodingPair {α : Type u_1} {β : Type u_2} {Γ₁ : Type u_3} {Γ₂ : Type u_4} [Fintype Γ₁] [Fintype Γ₂] (ea : Encoding α Γ₁) (eb : Encoding β Γ₂) : Encoding (α × β) (Γ₁ ⊕ Γ₂)

Deprecated alias for encodingProd.

@[deprecated Computability.Encoding.card_le_aleph0 (since := "2026-05-07")]

theorem Computability.FinEncoding.card_le_aleph0 {α : Type u_1} {Γ : Type u_2} [Countable Γ] (e : Encoding α Γ) : Cardinal.mk α ≤ Cardinal.aleph0

Deprecated alias for Encoding.card_le_aleph0.