Gödel Numbering for Partial Recursive Functions.

This file defines Nat.Partrec.Code, an inductive datatype describing code for partial recursive functions on ℕ. It defines an encoding for these codes, and proves that the constructors are primitive recursive with respect to the encoding.

It also defines the evaluation of these codes as partial functions using PFun, and proves that a function is partially recursive (as defined by Nat.Partrec) if and only if it is the evaluation of some code.

Main Definitions

• Nat.Partrec.Code: Inductive datatype for partial recursive codes.
• Nat.Partrec.Code.encodeCode: A (computable) encoding of codes as natural numbers.
• Nat.Partrec.Code.ofNatCode: The inverse of this encoding.
• Nat.Partrec.Code.eval: The interpretation of a Nat.Partrec.Code as a partial function.

Main Results

• Nat.Partrec.Code.primrec_recOn: Recursion on Nat.Partrec.Code is primitive recursive.
• Nat.Partrec.Code.computable_recOn: Recursion on Nat.Partrec.Code is computable.
• Nat.Partrec.Code.smn: The $S_n^m$ theorem.
• Nat.Partrec.Code.exists_code: Partial recursiveness is equivalent to being the eval of a code.
• Nat.Partrec.Code.primrec_evaln: evaln is primitive recursive.
• Nat.Partrec.Code.fixed_point: Roger's fixed point theorem.
• Nat.Partrec.Code.fixed_point₂: Kleene's second recursion theorem.

References

• [Mario Carneiro, Formalizing computability theory via partial recursive functions][carneiro2019]

theorem Nat.Partrec.rfind' {f : ℕ →. ℕ} (hf : Nat.Partrec f) : Nat.Partrec (unpaired fun (a m : ℕ) => Part.map (fun (x : ℕ) => x + m) (Nat.rfind fun (n : ℕ) => (fun (m : ℕ) => decide (m = 0)) <$> f (Nat.pair a (n + m))))

inductive Nat.Partrec.Code : Type

Code for partial recursive functions from ℕ to ℕ. See Nat.Partrec.Code.eval for the interpretation of these constructors.

• zero : Code
• succ : Code
• left : Code
• right : Code
• pair : Code → Code → Code
• comp : Code → Code → Code
• prec : Code → Code → Code
• rfind' : Code → Code

@[instance_reducible]

instance Nat.Partrec.Code.instInhabited : Inhabited Code

Equations

• Nat.Partrec.Code.instInhabited = { default := Nat.Partrec.Code.zero }

def Nat.Partrec.Code.const : ℕ → Code

Returns a code for the constant function outputting a particular natural.

Equations

• Nat.Partrec.Code.const 0 = Nat.Partrec.Code.zero
• Nat.Partrec.Code.const n.succ = Nat.Partrec.Code.succ.comp (Nat.Partrec.Code.const n)

theorem Nat.Partrec.Code.const_inj {n₁ n₂ : ℕ} : Code.const n₁ = Code.const n₂ → n₁ = n₂

def Nat.Partrec.Code.id : Code

A code for the identity function.

Equations

• Nat.Partrec.Code.id = Nat.Partrec.Code.left.pair Nat.Partrec.Code.right

def Nat.Partrec.Code.curry (c : Code) (n : ℕ) : Code

Given a code c taking a pair as input, returns a code using n as the first argument to c.

Equations

• c.curry n = c.comp ((Nat.Partrec.Code.const n).pair Nat.Partrec.Code.id)

def Nat.Partrec.Code.encodeCode : Code → ℕ

An encoding of a Nat.Partrec.Code as a ℕ.

Equations

• Nat.Partrec.Code.zero.encodeCode = 0
• Nat.Partrec.Code.succ.encodeCode = 1
• Nat.Partrec.Code.left.encodeCode = 2
• Nat.Partrec.Code.right.encodeCode = 3
• (cf.pair cg).encodeCode = 2 * (2 * Nat.pair cf.encodeCode cg.encodeCode) + 4
• (cf.comp cg).encodeCode = 2 * (2 * Nat.pair cf.encodeCode cg.encodeCode + 1) + 4
• (cf.prec cg).encodeCode = 2 * (2 * Nat.pair cf.encodeCode cg.encodeCode) + 1 + 4
• cf.rfind'.encodeCode = 2 * (2 * cf.encodeCode + 1) + 1 + 4

@[irreducible]

def Nat.Partrec.Code.ofNatCode : ℕ → Code

A decoder for Nat.Partrec.Code.encodeCode, taking any ℕ to the Nat.Partrec.Code it represents.

Equations

• One or more equations did not get rendered due to their size.
• Nat.Partrec.Code.ofNatCode 0 = Nat.Partrec.Code.zero
• Nat.Partrec.Code.ofNatCode 1 = Nat.Partrec.Code.succ
• Nat.Partrec.Code.ofNatCode 2 = Nat.Partrec.Code.left
• Nat.Partrec.Code.ofNatCode 3 = Nat.Partrec.Code.right

@[instance_reducible]

instance Nat.Partrec.Code.instDenumerable : Denumerable Code

Equations

• One or more equations did not get rendered due to their size.

theorem Nat.Partrec.Code.encodeCode_eq : Encodable.encode = encodeCode

theorem Nat.Partrec.Code.ofNatCode_eq : Denumerable.ofNat Code = ofNatCode

theorem Nat.Partrec.Code.encode_lt_pair (cf cg : Code) : Encodable.encode cf < Encodable.encode (cf.pair cg) ∧ Encodable.encode cg < Encodable.encode (cf.pair cg)

theorem Nat.Partrec.Code.encode_lt_comp (cf cg : Code) : Encodable.encode cf < Encodable.encode (cf.comp cg) ∧ Encodable.encode cg < Encodable.encode (cf.comp cg)

theorem Nat.Partrec.Code.encode_lt_prec (cf cg : Code) : Encodable.encode cf < Encodable.encode (cf.prec cg) ∧ Encodable.encode cg < Encodable.encode (cf.prec cg)

theorem Nat.Partrec.Code.encode_lt_rfind' (cf : Code) : Encodable.encode cf < Encodable.encode cf.rfind'

theorem Nat.Partrec.Code.primrec₂_pair : Primrec₂ pair

theorem Nat.Partrec.Code.primrec₂_comp : Primrec₂ comp

theorem Nat.Partrec.Code.primrec₂_prec : Primrec₂ prec

theorem Nat.Partrec.Code.primrec_rfind' : Primrec rfind'

theorem Nat.Partrec.Code.primrec_recOn' {α : Type u_1} {σ : Type u_2} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ} (hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ} (hr : Primrec r) {pr : α → Code × Code × σ × σ → σ} (hpr : Primrec₂ pr) {co : α → Code × Code × σ × σ → σ} (hco : Primrec₂ co) {pc : α → Code × Code × σ × σ → σ} (hpc : Primrec₂ pc) {rf : α → Code × σ → σ} (hrf : Primrec₂ rf) : have PR := fun (a : α) (cf cg : Code) (hf hg : σ) => pr a (cf, cg, hf, hg); have CO := fun (a : α) (cf cg : Code) (hf hg : σ) => co a (cf, cg, hf, hg); have PC := fun (a : α) (cf cg : Code) (hf hg : σ) => pc a (cf, cg, hf, hg); have RF := fun (a : α) (cf : Code) (hf : σ) => rf a (cf, hf); have F := fun (a : α) (c : Code) => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a); Primrec fun (a : α) => F a (c a)

theorem Nat.Partrec.Code.primrec_recOn {α : Type u_1} {σ : Type u_2} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ} (hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ} (hr : Primrec r) {pr : α → Code → Code → σ → σ → σ} (hpr : Primrec fun (a : α × Code × Code × σ × σ) => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2) {co : α → Code → Code → σ → σ → σ} (hco : Primrec fun (a : α × Code × Code × σ × σ) => co a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2) {pc : α → Code → Code → σ → σ → σ} (hpc : Primrec fun (a : α × Code × Code × σ × σ) => pc a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2) {rf : α → Code → σ → σ} (hrf : Primrec fun (a : α × Code × σ) => rf a.1 a.2.1 a.2.2) : have F := fun (a : α) (c : Code) => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a); Primrec fun (a : α) => F a (c a)

Recursion on Nat.Partrec.Code is primitive recursive.

theorem Nat.Partrec.Code.computable_recOn {α : Type u_1} {σ : Type u_2} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c) {z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l) {r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ} (hpr : Computable₂ pr) {co : α → Code × Code × σ × σ → σ} (hco : Computable₂ co) {pc : α → Code × Code × σ × σ → σ} (hpc : Computable₂ pc) {rf : α → Code × σ → σ} (hrf : Computable₂ rf) : have PR := fun (a : α) (cf cg : Code) (hf hg : σ) => pr a (cf, cg, hf, hg); have CO := fun (a : α) (cf cg : Code) (hf hg : σ) => co a (cf, cg, hf, hg); have PC := fun (a : α) (cf cg : Code) (hf hg : σ) => pc a (cf, cg, hf, hg); have RF := fun (a : α) (cf : Code) (hf : σ) => rf a (cf, hf); have F := fun (a : α) (c : Code) => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a); Computable fun (a : α) => F a (c a)

Recursion on Nat.Partrec.Code is computable.

def Nat.Partrec.Code.eval : Code → ℕ →. ℕ

The interpretation of a Nat.Partrec.Code as a partial function.

• Nat.Partrec.Code.zero: The constant zero function.
• Nat.Partrec.Code.succ: The successor function.
• Nat.Partrec.Code.left: Left unpairing of a pair of ℕ (encoded by Nat.pair)
• Nat.Partrec.Code.right: Right unpairing of a pair of ℕ (encoded by Nat.pair)
• Nat.Partrec.Code.pair: Pairs the outputs of argument codes using Nat.pair.
• Nat.Partrec.Code.comp: Composition of two argument codes.
• Nat.Partrec.Code.prec: Primitive recursion. Given an argument of the form Nat.pair a n:
  • If n = 0, returns eval cf a.
  • If n = succ k, returns eval cg (pair a (pair k (eval (prec cf cg) (pair a k))))
• Nat.Partrec.Code.rfind': Minimization starting at a provided value. Given an argument of the form Nat.pair a m, returns the least n ≥ m such that eval cf (pair a n) = 0, if such an n exists and if eval cf (pair a k) terminates for all m ≤ k ≤ n.

Equations

• Nat.Partrec.Code.zero.eval = pure 0
• Nat.Partrec.Code.succ.eval = ↑Nat.succ
• Nat.Partrec.Code.left.eval = ↑fun (n : ℕ) => (Nat.unpair n).1
• Nat.Partrec.Code.right.eval = ↑fun (n : ℕ) => (Nat.unpair n).2
• (cf.pair cg).eval = fun (n : ℕ) => Nat.pair <$> cf.eval n <*> cg.eval n
• (cf.comp cg).eval = fun (n : ℕ) => cg.eval n >>= cf.eval
• (cf.prec cg).eval = Nat.unpaired fun (a n : ℕ) => Nat.rec (cf.eval a) (fun (y : ℕ) (IH : Part ℕ) => do let i ← IH cg.eval (Nat.pair a (Nat.pair y i))) n
• cf.rfind'.eval = Nat.unpaired fun (a m : ℕ) => Part.map (fun (x : ℕ) => x + m) (Nat.rfind fun (n : ℕ) => (fun (m : ℕ) => decide (m = 0)) <$> cf.eval (Nat.pair a (n + m)))

@[simp]

theorem Nat.Partrec.Code.eval_prec_zero (cf cg : Code) (a : ℕ) : (cf.prec cg).eval (Nat.pair a 0) = cf.eval a

Helper lemma for the evaluation of prec in the base case.

theorem Nat.Partrec.Code.eval_prec_succ (cf cg : Code) (a k : ℕ) : (cf.prec cg).eval (Nat.pair a k.succ) = do let ih ← (cf.prec cg).eval (Nat.pair a k) cg.eval (Nat.pair a (Nat.pair k ih))

Helper lemma for the evaluation of prec in the recursive case.

@[instance_reducible]

instance Nat.Partrec.Code.instMembershipPFun : Membership (ℕ →. ℕ) Code

Equations

• Nat.Partrec.Code.instMembershipPFun = { mem := fun (c : Nat.Partrec.Code) (f : ℕ →. ℕ) => c.eval = f }

@[simp]

theorem Nat.Partrec.Code.eval_const (n m : ℕ) : (Code.const n).eval m = Part.some n

@[simp]

theorem Nat.Partrec.Code.eval_id (n : ℕ) : Code.id.eval n = Part.some n

@[simp]

theorem Nat.Partrec.Code.eval_curry (c : Code) (n x : ℕ) : (c.curry n).eval x = c.eval (Nat.pair n x)

theorem Nat.Partrec.Code.primrec_const : Primrec Code.const

theorem Nat.Partrec.Code.primrec₂_curry : Primrec₂ curry

theorem Nat.Partrec.Code.curry_inj {c₁ c₂ : Code} {n₁ n₂ : ℕ} (h : c₁.curry n₁ = c₂.curry n₂) : c₁ = c₂ ∧ n₁ = n₂

theorem Nat.Partrec.Code.smn : ∃ (f : Code → ℕ → Code), Computable₂ f ∧ ∀ (c : Code) (n x : ℕ), (f c n).eval x = c.eval (Nat.pair n x)

The $S_n^m$ theorem: There is a computable function, namely Nat.Partrec.Code.curry, that takes a program and a ℕ n, and returns a new program using n as the first argument.

theorem Nat.Partrec.Code.exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ (c : Code), c.eval = f

A function is partial recursive if and only if there is a code implementing it. Therefore, eval is a universal partial recursive function.

@[irreducible]

def Nat.Partrec.Code.evaln : ℕ → Code → ℕ → Option ℕ

A modified evaluation for the code which returns an Option ℕ instead of a Part ℕ. To avoid undecidability, evaln takes a parameter k and fails if it encounters a number ≥ k in the course of its execution. Other than this, the semantics are the same as in Nat.Partrec.Code.eval.

Equations

• One or more equations did not get rendered due to their size.
• Nat.Partrec.Code.evaln 0 x✝ = fun (x : ℕ) => none
• Nat.Partrec.Code.evaln k.succ Nat.Partrec.Code.zero = fun (n : ℕ) => do guard (n ≤ k) pure 0
• Nat.Partrec.Code.evaln k.succ Nat.Partrec.Code.succ = fun (n : ℕ) => do guard (n ≤ k) pure n.succ
• Nat.Partrec.Code.evaln k.succ Nat.Partrec.Code.left = fun (n : ℕ) => do guard (n ≤ k) pure (Nat.unpair n).1
• Nat.Partrec.Code.evaln k.succ Nat.Partrec.Code.right = fun (n : ℕ) => do guard (n ≤ k) pure (Nat.unpair n).2
• Nat.Partrec.Code.evaln k.succ (cf.pair cg) = fun (n : ℕ) => do guard (n ≤ k) Nat.pair <$> Nat.Partrec.Code.evaln (k + 1) cf n <*> Nat.Partrec.Code.evaln (k + 1) cg n
• Nat.Partrec.Code.evaln k.succ (cf.comp cg) = fun (n : ℕ) => do guard (n ≤ k) let x ← Nat.Partrec.Code.evaln (k + 1) cg n Nat.Partrec.Code.evaln (k + 1) cf x

theorem Nat.Partrec.Code.evaln_bound {k : ℕ} {c : Code} {n x : ℕ} : x ∈ evaln k c n → n < k

theorem Nat.Partrec.Code.evaln_mono {k₁ k₂ : ℕ} {c : Code} {n x : ℕ} : k₁ ≤ k₂ → x ∈ evaln k₁ c n → x ∈ evaln k₂ c n

theorem Nat.Partrec.Code.evaln_sound {k : ℕ} {c : Code} {n x : ℕ} : x ∈ evaln k c n → x ∈ c.eval n

theorem Nat.Partrec.Code.evaln_complete {c : Code} {n x : ℕ} : x ∈ c.eval n ↔ ∃ (k : ℕ), x ∈ evaln k c n

theorem Nat.Partrec.Code.primrec_evaln : Primrec fun (a : (ℕ × Code) × ℕ) => evaln a.1.1 a.1.2 a.2

The Nat.Partrec.Code.evaln function is primitive recursive.

theorem Nat.Partrec.Code.eval_eq_rfindOpt (c : Code) (n : ℕ) : c.eval n = rfindOpt fun (k : ℕ) => evaln k c n

theorem Nat.Partrec.Code.eval_part : Partrec₂ eval

theorem Nat.Partrec.Code.fixed_point {f : Code → Code} (hf : Computable f) : ∃ (c : Code), (f c).eval = c.eval

Roger's fixed-point theorem: any total, computable f has a fixed point. That is, under the interpretation given by Nat.Partrec.Code.eval, there is a code c such that c and f c have the same evaluation.

theorem Nat.Partrec.Code.fixed_point₂ {f : Code → ℕ →. ℕ} (hf : Partrec₂ f) : ∃ (c : Code), c.eval = f c

Kleene's second recursion theorem

instance Nat.Partrec.Code.instCountableSubtypePFunPartrec : Countable { f : ℕ →. ℕ // Partrec f }

There are only countably many partial recursive partial functions ℕ →. ℕ.

instance Nat.Partrec.Code.instCountableSubtypeForallComputable : Countable { f : ℕ → ℕ // Computable f }

There are only countably many computable functions ℕ → ℕ.