openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentNonidentity.lean
| import OAI.Computability.DegreeRigidity.Computability.ArithmeticCoanalytic | |
| namespace OAI | |
| namespace TuringRigidity.ArithmeticPersistence | |
| open EncodedForcing OracleJump UniformArithmetic PersistentRestrictions | |
| open PersistentPresentation PersistentCountability NumericalAutomorphism | |
| noncomputable section | |
| def Moved (A R : Oracle) : Prop := | |
| ∃ n m, R (Nat.pair n m) = true ∧ degree (columns A n) ≠ degree (columns A m) | |
| theorem moved_iff {I : CountableIdeal} {A R : Oracle} (hA : Presented I A) | |
| (ρ : I ≃o I) (hR : ∀ v, R v = true ↔ Graph ρ hA v) : | |
| Moved A R ↔ ∃ x : I, ρ x ≠ x := by | |
| constructor | |
| · rintro ⟨n,m,hr,hne⟩ | |
| refine ⟨entry hA n,fun he => ?_⟩ | |
| have hg := (hR (Nat.pair n m)).mp hr | |
| simp only [PersistentPresentation.Graph,Nat.unpair_pair] at hg | |
| exact hne ((congrArg Subtype.val he).symm.trans hg) | |
| · rintro ⟨x,hx⟩ | |
| obtain ⟨n,hn⟩ := (hA x.val).mp x.property | |
| obtain ⟨m,hm⟩ := (hA (ρ x).val).mp (ρ x).property | |
| have he : entry hA n = x := Subtype.ext hn | |
| refine ⟨n,m,(hR _).mpr ?_,?_⟩ | |
| · simp only [PersistentPresentation.Graph,Nat.unpair_pair] | |
| rw [he,hm] | |
| · intro hh | |
| apply hx | |
| apply Subtype.ext | |
| exact hm.symm.trans (hh.symm.trans hn) | |
| theorem moved_arith : Arith (fun O (_ : ℕ) => Moved (O 0) (O 1)) := by | |
| have hp := query_at (parameter_arith 1) (Primrec.const 0) | |
| (Primrec₂.natPair.comp (UniformArithmetic.right_primrec.comp UniformArithmetic.left_primrec) | |
| UniformArithmetic.right_primrec) | |
| have he := degree_equal_arith | |
| (column_arith (parameter_arith 0) (UniformArithmetic.right_primrec.comp UniformArithmetic.left_primrec)) | |
| (column_arith (parameter_arith 0) UniformArithmetic.right_primrec) | |
| exact (hp.and he.neg).ex.ex.congr | |
| (fun _ _ => by simp only [Moved,UniformArithmetic.left,UniformArithmetic.right,Nat.unpair_pair]) | |
| theorem nonidentity_piOneOne : PiOneOne (fun A R => Property A R ∧ Moved A R) := by | |
| obtain ⟨Q,hQ,hiff⟩ := source_4_2_1 | |
| refine ⟨fun O v => Q O v ∧ Moved (O 0) (O 1),hQ.and moved_arith,fun A R => ?_⟩ | |
| dsimp only | |
| rw [hiff] | |
| change ((∀ H, Q (parameters A R H) 0) ∧ Moved A R) ↔ | |
| ∀ H, Q (parameters A R H) 0 ∧ Moved A R | |
| exact ⟨fun h H => ⟨h.1 H,h.2⟩,fun h => ⟨fun H => (h H).1,(h FixedArithmetic.zero).2⟩⟩ | |
| theorem nonidentity_coanalytic : MeasureTheory.AnalyticSet | |
| ({x : Oracle × Oracle | Property x.1 x.2 ∧ Moved x.1 x.2}ᶜ) := | |
| nonidentity_piOneOne.coanalytic | |
| theorem nonrigidity_witness (h : ¬ MainTheorem) : | |
| ∃ A R : Oracle, Property A R ∧ Moved A R := by | |
| classical | |
| simp only [MainTheorem,not_forall] at h | |
| obtain ⟨π,x,hx⟩ := h | |
| let z := degree (jump FixedArithmetic.zero) | |
| obtain ⟨I,hi,hf,hb⟩ := countable_invariant_hull π {x,z} (Set.toFinite _).countable | |
| let ρ := restrict π I hf hb | |
| have hρ : RestrictsTo π I ρ := fun _ => rfl | |
| have hp : Persistent I ρ := persistent_of_global_restriction π I ρ hρ | |
| have hz : z ∈ I.carrier := hi (by simp) | |
| have hxi : x ∈ I.carrier := hi (by simp) | |
| have hn : ∃ a : I, ρ a ≠ a := by | |
| refine ⟨⟨x,hxi⟩,fun he => ?_⟩ | |
| exact hx (congrArg Subtype.val he) | |
| obtain ⟨A,hA⟩ := presentation_exists I | |
| let R := graphOracle hA ρ | |
| have hR : ∀ v, R v = true ↔ Graph ρ hA v := fun v => by simp [R,graphOracle] | |
| exact ⟨A,R,⟨I,hA,hz,ρ,hR,hp⟩,(moved_iff hA ρ hR).mpr hn⟩ | |
| end | |
| end TuringRigidity.ArithmeticPersistence | |
| end OAI | |