openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /EnumerationConstruction.lean
| 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 | |