openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CountedRealsPresentation.lean
| import OAI.Computability.DegreeRigidity.SetModels.SetModelArithmeticDischarge | |
| import OAI.Computability.DegreeRigidity.SetModels.CountableSetReal | |
| namespace OAI | |
| namespace TuringRigidity.SetModelCountability | |
| open BoundedSetTheory TransitiveNameModel SetModelReals SetModelFunctions | |
| open SetModelSequences SetDegreeDecoding PersistentRestrictions PersistentPresentation | |
| open EncodedForcing SetModelArithmetic | |
| universe u | |
| theorem counted_reals_columns {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M) | |
| (hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x) | |
| (hcount : BoundedSetTheory.InternallyCountable M R) : | |
| ∃ A ∈ reals M, ∀ B : Oracle, (∃ n, columns A n = B) ↔ realSet B ∈ R := by | |
| classical | |
| obtain ⟨q,hq,hqf,hqo⟩ := hcount | |
| have hex (n : ℕ) : ∃ B : Oracle, realSet B ∈ R ∧ | |
| ZFSet.pair (natSet n) (realSet B) ∈ q := by | |
| obtain ⟨x,hx,hnx,_⟩ := hqf.2 (natSet n) ((mem_omega _).mpr ⟨n,rfl⟩) | |
| obtain ⟨B,rfl⟩ := hreal x hx | |
| exact ⟨B,hx,hnx⟩ | |
| choose B hB using hex | |
| have hseq : q = sequenceSet B := by | |
| apply ZFSet.ext; intro z | |
| constructor | |
| · intro hz | |
| obtain ⟨x,hx,y,_,rfl⟩ := hqf.1 z hz | |
| obtain ⟨n,rfl⟩ := (mem_omega x).mp hx | |
| have hy := hqf.functional ((mem_omega _).mpr ⟨n,rfl⟩) hz (hB n).2 | |
| exact (mem_sequenceSet B _).mpr ⟨n,by rw [hy]⟩ | |
| · intro hz | |
| obtain ⟨n,rfl⟩ := (mem_sequenceSet B z).mp hz | |
| exact (hB n).2 | |
| have hBM (n : ℕ) : B n ∈ reals M := C.transitive R hR _ (hB n).1 | |
| obtain ⟨r,hr,hrc⟩ := internal_reals M C.transitive C.power C.separation C.infinity | |
| let A : Oracle := fun v => B (Nat.unpair v).1 (Nat.unpair v).2 | |
| have hAM : A ∈ reals M := flatten_mem M C.transitive C.pairing C.union C.power | |
| C.separation C.infinity hr hrc C.pairing_mem pairingSet_code B hBM (hseq ▸ hq) | |
| have hcol (n : ℕ) : columns A n = B n := by | |
| funext k; simp only [columns,A,Nat.unpair_pair] | |
| refine ⟨A,hAM,fun X => ?_⟩ | |
| constructor | |
| · rintro ⟨n,rfl⟩; rw [hcol]; exact (hB n).1 | |
| · intro hX | |
| obtain ⟨n,hn,hnX⟩ := hqo _ hX | |
| obtain ⟨n,rfl⟩ := (mem_omega n).mp hn | |
| have he := hqf.functional ((mem_omega _).mpr ⟨n,rfl⟩) (hB n).2 hnX | |
| refine ⟨n,?_⟩ | |
| rw [hcol] | |
| exact realSet_injective he | |
| theorem presentation_of_counted_reals {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M) | |
| (hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x) | |
| (hcount : BoundedSetTheory.InternallyCountable M R) (I : CountableIdeal) | |
| (hI : ∀ d, d ∈ I.carrier ↔ ∃ B : Oracle, realSet B ∈ R ∧ degree B = d) : | |
| ∃ A ∈ reals M, Presented I A := by | |
| obtain ⟨A,hA,hcols⟩ := counted_reals_columns C hR hreal hcount | |
| refine ⟨A,hA,fun d => ?_⟩ | |
| rw [hI] | |
| constructor | |
| · rintro ⟨B,hB,hBd⟩ | |
| obtain ⟨n,hn⟩ := (hcols B).mpr hB | |
| exact ⟨n,by rw [hn,hBd]⟩ | |
| · rintro ⟨n,hn⟩ | |
| exact ⟨columns A n,(hcols _).mp ⟨n,rfl⟩,hn⟩ | |
| theorem persistent_mem_of_counted_reals {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M) | |
| (hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x) | |
| (hcount : BoundedSetTheory.InternallyCountable M R) (I : CountableIdeal) | |
| (hI : ∀ d, d ∈ I.carrier ↔ ∃ B : Oracle, realSet B ∈ R ∧ degree B = d) | |
| (ρ : I ≃o I) (hρ : Persistent I ρ) | |
| (hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier) : | |
| automorphismSet ρ ∈ M := by | |
| obtain ⟨A,hAM,hA⟩ := presentation_of_counted_reals C hR hreal hcount I hI | |
| exact (SetModelSyntax.persistent_sets C hA hAM ρ hρ hz).2 | |
| end TuringRigidity.SetModelCountability | |
| end OAI | |