| import OAI.MathematicalPhysics.RapidForcing.RationalProfiles |
| import OAI.MathematicalPhysics.RapidForcing.ArithmeticAutomation |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Encodable Denumerable |
| namespace RapidForcing.EffectiveArithmetic |
|
|
| @[fun_prop] lemma primrec_nat_coprime : PrimrecPred (fun p : ℕ × ℕ => p.1.Coprime p.2) := by |
| simpa only [Nat.Coprime] using |
| (Primrec.eq.comp primrec_nat_gcd (Primrec.const 1)) |
|
|
| lemma rat_encoding (q : ℚ) : encode q = encode (q.num, q.den) := rfl |
|
|
| lemma rat_roundtrip (n : ℕ) : encode (decode (α := ℚ) n) = |
| encode ((decode (α := ℤ × ℕ) n).bind fun p => |
| if 0 < p.2 ∧ p.1.natAbs.Coprime p.2 then some (encode p) else none) := by |
| simp only [Encodable.decode_ofEquiv, Encodable.decode_sigma_val, |
| Encodable.decode_nat, Option.map_some, Option.bind_some, Option.map_map] |
| simp only [Encodable.decode, Encodable.decodeSubtype, Option.bind_some] |
| dsimp only [Equiv.symm_symm, Equiv.sigmaEquivProd, Equiv.coe_fn_mk, Function.comp_def] |
| split_ifs with h <;> |
| simp_all [Encodable.encode_ofEquiv, Encodable.Subtype.encode_eq, Nat.Coprime] |
| split_ifs with hh |
| · exact False.elim (h hh.1 hh.2) |
| · rfl |
|
|
| instance ratPrimcodable : Primcodable ℚ := |
| { toEncodable := Rat.instEncodable |
| prim := by |
| have hg : Primrec (fun p : ℤ × ℕ => |
| if 0 < p.2 ∧ p.1.natAbs.Coprime p.2 then some (encode p) else none) := by |
| fun_prop |
| have hb : Primrec (fun n : ℕ => (decode (α := ℤ × ℕ) n).bind fun p => |
| if 0 < p.2 ∧ p.1.natAbs.Coprime p.2 then some (encode p) else none) := |
| Primrec.option_bind Primrec.decode (hg.comp Primrec.snd).to₂ |
| exact (Primrec.nat_iff.mp (Primrec.encode.comp hb)).of_eq (fun n => (rat_roundtrip n).symm) } |
|
|
| @[fun_prop] lemma primrec_rat_num : Primrec Rat.num := by |
| have hp : Primrec (fun q : ℚ => (decode (α := ℤ × ℕ) (encode q)).getD (0, 0)) := |
| (Primrec.option_getD.comp (Primrec.decode.comp Primrec.encode) (Primrec.const (0, 0))) |
| exact (Primrec.fst.comp hp).of_eq fun q => by rw [rat_encoding, Encodable.encodek]; rfl |
|
|
| @[fun_prop] lemma primrec_rat_den : Primrec Rat.den := by |
| have hp : Primrec (fun q : ℚ => (decode (α := ℤ × ℕ) (encode q)).getD (0, 0)) := |
| (Primrec.option_getD.comp (Primrec.decode.comp Primrec.encode) (Primrec.const (0, 0))) |
| exact (Primrec.snd.comp hp).of_eq fun q => by rw [rat_encoding, Encodable.encodek]; rfl |
|
|
| end RapidForcing.EffectiveArithmetic |
|
|
| end |
|
|
| end OAI |
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|
