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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import OAI.Computability.DegreeRigidity.SetModels.EnumerationDecoding
import OAI.Computability.DegreeRigidity.SetModels.FixedArithmetic
namespace OAI
namespace TuringRigidity.EnumerationConstruction
open Encodable EncodedForcing ArithmeticHierarchy UniformOracle OracleJump
open ArithmeticModelDecoding SetCoding RelationCoding BoundedDecoding EnumerationDecoding
def rows (C H : Oracle) : Oracle := fun v =>
if (Nat.unpair (Nat.unpair v).1).1 = 0 then C (Nat.pair (Nat.unpair (Nat.unpair v).1).2 (Nat.unpair v).2)
else H (Nat.pair (Nat.unpair (Nat.unpair v).1).2 (Nat.unpair v).2)
theorem rows_reduces (C H : Oracle) : Reduces (rows C H) (join C H) := by
let f := Primrec.fst.comp Primrec.unpair
let r := Primrec.snd.comp Primrec.unpair
let p := Primrec₂.natPair.comp (r.comp f) r
let even := Primrec.nat_mul.comp (Primrec.const 2) p
have hi := Primrec.ite (Primrec.eq.comp (f.comp f) (Primrec.const 0)) even (Primrec.succ.comp even)
have hh := EffectiveFamilies.reindex_reduces (join C H) hi
convert hh using 1
funext v
by_cases h : (Nat.unpair (Nat.unpair v).1).1 = 0
· simp only [rows,ite_eq_left h,join_even]
· simp only [rows,ite_eq_right h,join_odd]
@[simp] theorem row_zero (C H : Oracle) (k : ℕ) : columns (rows C H) (Nat.pair 0 k) = columns C k := by
funext n; simp [columns,rows]
@[simp] theorem row_one (C H : Oracle) (k : ℕ) : columns (rows C H) (Nat.pair 1 k) = columns H k := by
funext n; simp [columns,rows]
theorem payload_code {C H : Oracle} (hC : Reduces C H) : ∃ p : RelationCode 2,
RelationBelow p (degree (jump H)) ∧ ∀ a b,
p.Holds (two a b) ↔ ∃ k, a = degree (columns C k) ∧ b = degree (columns H k) := by
have hP : RecursivePred (rows C H) (fun _ => True) := by
exact (total_primrec (Primrec.const 1)).of_eq (fun n => by simp)
obtain ⟨p,hp,hcode⟩ := EffectivePresentation.effective_coding (rows C H) (fun _ => True) hP 2
have hb := reduces_trans (rows_reduces C H) (join_reduces hC (reduces_refl H))
refine ⟨p,hp.mono (jump_mono hb),fun a b => ?_⟩
rw [hcode]
constructor
· rintro ⟨k,_,hk⟩
exact ⟨k,by simpa [two] using (hk 0).symm,by simpa [two] using (hk 1).symm⟩
· rintro ⟨k,rfl,rfl⟩
refine ⟨k,trivial,fun i => ?_⟩
fin_cases i <;> simp [two]
theorem enumeration_exists (H : Oracle) (hz : Reduces (jump FixedArithmetic.zero) H) :
∃ c : Enumeration, c.output = (fun k => degree (columns H k)) ∧
RelationBelow c.successor (degree (jump H)) ∧ RelationBelow c.payload (degree (jump H)) := by
obtain ⟨K⟩ := FixedArithmetic.frame_exists
have hC : Reduces K.C H := reduces_trans (reduces_jump K.C) (reduces_trans K.low hz)
obtain ⟨P,hP,hspec⟩ := payload_code hC
let c : Enumeration := ⟨fun k => degree (columns K.C k),K.injective,
fun k => degree (columns H k),K.successor,P,K.spec,hspec⟩
exact ⟨c,rfl,K.below.mono (reduces_trans hz (reduces_jump H)),hP⟩
end TuringRigidity.EnumerationConstruction
end OAI
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