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computability.partrec_code

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 evalution 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 #

Main Results #

References #

theorem nat.partrec.rfind' {f : →. } (hf : nat.partrec f) :
nat.partrec (nat.unpaired (λ (a m : ), part.map (λ (_x : ), _x + m) (nat.rfind (λ (n : ), (λ (m : ), decidable.to_bool (m = 0)) <$> f (nat.mkpair a (n + m))))))
inductive nat.partrec.code  :

Code for partial recursive functions from ℕ to ℕ. See nat.partrec.code.eval for the interpretation of these constructors.

Instances for nat.partrec.code
@[protected]

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

Equations
@[protected]

A code for the identity function.

Equations

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

Equations

A decoder for nat.partrec.code.encode_code, taking any ℕ to the nat.partrec.code it represents.

Equations
@[protected, instance]
Equations
theorem nat.partrec.code.rec_prim' {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] {c : α nat.partrec.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 nat.partrec.code × nat.partrec.code × σ × σ σ} (hpr : primrec₂ pr) {co : α nat.partrec.code × nat.partrec.code × σ × σ σ} (hco : primrec₂ co) {pc : α nat.partrec.code × nat.partrec.code × σ × σ σ} (hpc : primrec₂ pc) {rf : α nat.partrec.code × σ σ} (hrf : primrec₂ rf) :
let PR : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), pr a (cf, cg, hf, hg), CO : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), co a (cf, cg, hf, hg), PC : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), pc a (cf, cg, hf, hg), RF : α nat.partrec.code σ σ := λ (a : α) (cf : nat.partrec.code) (hf : σ), rf a (cf, hf), F : α nat.partrec.code σ := λ (a : α) (c : nat.partrec.code), c.rec_on (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a) in primrec (λ (a : α), F a (c a))
theorem nat.partrec.code.rec_prim {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] {c : α nat.partrec.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 nat.partrec.code nat.partrec.code σ σ σ} (hpr : primrec (λ (a : α × nat.partrec.code × nat.partrec.code × σ × σ), pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd)) {co : α nat.partrec.code nat.partrec.code σ σ σ} (hco : primrec (λ (a : α × nat.partrec.code × nat.partrec.code × σ × σ), co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd)) {pc : α nat.partrec.code nat.partrec.code σ σ σ} (hpc : primrec (λ (a : α × nat.partrec.code × nat.partrec.code × σ × σ), pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd)) {rf : α nat.partrec.code σ σ} (hrf : primrec (λ (a : α × nat.partrec.code × σ), rf a.fst a.snd.fst a.snd.snd)) :
let F : α nat.partrec.code σ := λ (a : α) (c : nat.partrec.code), c.rec_on (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) in primrec (λ (a : α), F a (c a))

Recursion on nat.partrec.code is primitive recursive.

theorem nat.partrec.code.rec_computable {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] {c : α nat.partrec.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 nat.partrec.code × nat.partrec.code × σ × σ σ} (hpr : computable₂ pr) {co : α nat.partrec.code × nat.partrec.code × σ × σ σ} (hco : computable₂ co) {pc : α nat.partrec.code × nat.partrec.code × σ × σ σ} (hpc : computable₂ pc) {rf : α nat.partrec.code × σ σ} (hrf : computable₂ rf) :
let PR : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), pr a (cf, cg, hf, hg), CO : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), co a (cf, cg, hf, hg), PC : α nat.partrec.code nat.partrec.code σ σ σ := λ (a : α) (cf cg : nat.partrec.code) (hf hg : σ), pc a (cf, cg, hf, hg), RF : α nat.partrec.code σ σ := λ (a : α) (cf : nat.partrec.code) (hf : σ), rf a (cf, hf), F : α nat.partrec.code σ := λ (a : α) (c : nat.partrec.code), c.rec_on (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a) in computable (λ (a : α), F a (c a))

Recursion on nat.partrec.code is computable.

The interpretation of a nat.partrec.code as a partial function.

Equations
@[simp]
theorem nat.partrec.code.eval_prec_zero (cf cg : nat.partrec.code) (a : ) :
(cf.prec cg).eval (nat.mkpair 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 : nat.partrec.code) (a k : ) :
(cf.prec cg).eval (nat.mkpair a k.succ) = (cf.prec cg).eval (nat.mkpair a k) >>= λ (ih : ), cg.eval (nat.mkpair a (nat.mkpair k ih))

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

@[protected, instance]
Equations
@[simp]
theorem nat.partrec.code.eval_curry (c : nat.partrec.code) (n x : ) :
(c.curry n).eval x = c.eval (nat.mkpair n x)
theorem nat.partrec.code.curry_inj {c₁ c₂ : nat.partrec.code} {n₁ n₂ : } (h : c₁.curry n₁ = c₂.curry n₂) :
c₁ = c₂ n₁ = n₂

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.

A function is partial recursive if and only if there is a code implementing it.

A modified evaluation for the code which returns an option instead of a part. 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
theorem nat.partrec.code.evaln_mono {k₁ k₂ : } {c : nat.partrec.code} {n x : } :

The nat.partrec.code.evaln function is primitive recursive.

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.