openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GenericIdentity.lean
| import OAI.Computability.DegreeRigidity.SetModels.IdealSelection | |
| import OAI.Computability.DegreeRigidity.Representation.GenericCodingBorel | |
| namespace OAI | |
| namespace TuringRigidity.GenericIdentity | |
| open Set | |
| def PrincipalIntersection (U L R : Oracle) : Prop := | |
| ∀ b : Degree, b ≤ degree U ↔ b ≤ degree L ∧ b ≤ degree R | |
| theorem principal_iff_countable (U L R : Oracle) : | |
| PrincipalIntersection U L R ↔ Reduces U L ∧ Reduces U R ∧ | |
| ∀ n, Reduces (programOutput (codeEnumeration n) L) L ∧ | |
| Reduces (programOutput (codeEnumeration n) L) R → | |
| Reduces (programOutput (codeEnumeration n) L) U := by | |
| constructor | |
| · intro h | |
| have hh := (h (degree U)).mp le_rfl | |
| exact ⟨hh.1,hh.2,fun n hn => (h (degree (programOutput (codeEnumeration n) L))).mpr hn⟩ | |
| · rintro ⟨hL,hR,h⟩ b | |
| obtain ⟨B,rfl⟩ := degree_surjective b | |
| constructor | |
| · intro hb | |
| exact ⟨reduces_trans hb hL,reduces_trans hb hR⟩ | |
| · rintro ⟨hbL,hbR⟩ | |
| obtain ⟨n,hn⟩ := exists_programOutput L B hbL | |
| have hh := h n (by simpa [hn] using And.intro hbL hbR) | |
| simpa [hn] using hh | |
| theorem principal_measurable {X : Type*} [MeasurableSpace X] | |
| (U L R : X → Oracle) (hU : Measurable U) (hL : Measurable L) (hR : Measurable R) : | |
| MeasurableSet {x | PrincipalIntersection (U x) (L x) (R x)} := by | |
| have hm : MeasurableSet {x | ∀ n, Reduces (programOutput (codeEnumeration n) (L x)) (L x) ∧ | |
| Reduces (programOutput (codeEnumeration n) (L x)) (R x) → | |
| Reduces (programOutput (codeEnumeration n) (L x)) (U x)} := by | |
| simp only [Set.ofPred_forall] | |
| apply MeasurableSet.iInter | |
| intro n | |
| let C : X → Oracle := fun x => programOutput (codeEnumeration n) (L x) | |
| have hC : Measurable C := (programOutput_measurable _).comp hL | |
| have he : {x | Reduces (C x) (L x) ∧ Reduces (C x) (R x) → Reduces (C x) (U x)} = | |
| ({x | Reduces (C x) (L x)} ∩ {x | Reduces (C x) (R x)})ᶜ ∪ {x | Reduces (C x) (U x)} := by | |
| ext x; simp only [mem_ofPred_eq,mem_union,mem_inter_iff,mem_compl_iff]; tauto | |
| change MeasurableSet {x | Reduces (C x) (L x) ∧ Reduces (C x) (R x) → Reduces (C x) (U x)} | |
| rw [he] | |
| exact ((reduces_measurable C L hC hL).inter (reduces_measurable C R hC hR)).compl.union | |
| (reduces_measurable C U hC hU) | |
| convert (reduces_measurable U L hU hL).inter ((reduces_measurable U R hU hR).inter hm) using 1 | |
| ext x | |
| simp only [mem_inter_iff,mem_ofPred_eq,principal_iff_countable] | |
| noncomputable def value (p : OracleCode) (P A : Oracle) : Oracle := programOutput p (join A P) | |
| def Total (p : OracleCode) (P A : Oracle) : Prop := | |
| ∃ B, OracleCode.eval (oracleFunction (join A P)) p = oracleFunction B | |
| theorem value_measurable (p : OracleCode) (P : Oracle) : Measurable (value p P) := | |
| (programOutput_measurable p).comp (join_measurable _ _ measurable_id measurable_const) | |
| theorem total_iff (p : OracleCode) (P A : Oracle) : | |
| Total p P A ↔ OracleCode.eval (oracleFunction (join A P)) p = oracleFunction (value p P A) := by | |
| constructor | |
| · rintro ⟨B,hB⟩ | |
| rw [show value p P A = B from programOutput_eq p _ B hB] | |
| exact hB | |
| · intro h; exact ⟨_,h⟩ | |
| theorem total_measurable (p : OracleCode) (P : Oracle) : MeasurableSet {A | Total p P A} := by | |
| simp only [total_iff] | |
| exact OracleCode.success_measurable p _ _ | |
| (join_measurable _ _ measurable_id measurable_const) (value_measurable p P) | |
| def SourceEquation (p : OracleCode) (P : Oracle) (z : Oracle × Oracle × Oracle) : Prop := | |
| Total p P z.1 ∧ Total p P z.2.1 ∧ Total p P z.2.2 ∧ | |
| Total p P (GenericCoding.code z.1 z.2.1) ∧ Total p P (GenericCoding.code z.1 z.2.2) ∧ | |
| PrincipalIntersection (value p P z.1) | |
| (join (value p P (GenericCoding.code z.1 z.2.1)) (value p P z.2.1)) | |
| (join (value p P (GenericCoding.code z.1 z.2.2)) (value p P z.2.2)) | |
| theorem sourceEquation_measurable (p : OracleCode) (P : Oracle) : | |
| MeasurableSet {z | SourceEquation p P z} := by | |
| let X : (Oracle × Oracle × Oracle) → Oracle := fun z => z.1 | |
| let G : (Oracle × Oracle × Oracle) → Oracle := fun z => z.2.1 | |
| let H : (Oracle × Oracle × Oracle) → Oracle := fun z => z.2.2 | |
| have hX : Measurable X := measurable_fst | |
| have hG : Measurable G := measurable_fst.comp measurable_snd | |
| have hH : Measurable H := measurable_snd.comp measurable_snd | |
| have hCG : Measurable (fun z => GenericCoding.code (X z) (G z)) := | |
| GenericCoding.code_measurable.comp (hX.prodMk hG) | |
| have hCH : Measurable (fun z => GenericCoding.code (X z) (H z)) := | |
| GenericCoding.code_measurable.comp (hX.prodMk hH) | |
| have hF := value_measurable p P | |
| have hI := principal_measurable _ _ _ (hF.comp hX) | |
| (join_measurable _ _ (hF.comp hCG) (hF.comp hG)) | |
| (join_measurable _ _ (hF.comp hCH) (hF.comp hH)) | |
| exact ((total_measurable p P).preimage hX).inter | |
| (((total_measurable p P).preimage hG).inter | |
| (((total_measurable p P).preimage hH).inter | |
| (((total_measurable p P).preimage hCG).inter | |
| (((total_measurable p P).preimage hCH).inter hI)))) | |
| end TuringRigidity.GenericIdentity | |
| end OAI | |