openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /LevelSubsetDefinition.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.LevelInlinedDefinition | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SetDegreeDecoding PersistentRestrictions | |
| universe u | |
| theorem Construction.subset_at_level (t : Construction.{u}) (M : ZFSet.{u}) | |
| (hM : Transitive M) (hT : SourceT M) (P : Oracle) | |
| (hP : P ∈ modelReals M) (ht : t.Certified (groundReals M) P) (hi : t.Indexed M) : | |
| ∃ γ : Ordinal.{u}, γ.toZFSet ∈ M ∧ | |
| (∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧ | |
| t.value (groundReals M) P = definedSubset (level (groundReals M) γ) t.singleMembership | |
| (fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i) := by | |
| obtain ⟨γ,hγ,hv,he,hf⟩ := t.inlined_at_level M hM hT P hP ht hi | |
| refine ⟨γ,hγ,he,?_⟩ | |
| apply ZFSet.ext; intro z | |
| rw [mem_definedSubset] | |
| have hs := t.singleMembership.finite_support (level (groundReals M) γ : Set ZFSet) | |
| (cons z (t.singleInputs (groundReals M) P)) | |
| (cons z (tupleEnv (fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i))) | |
| (by | |
| intro i hin | |
| cases i with | |
| | zero => rfl | |
| | succ i => | |
| simp only [cons_succ,tupleEnv,dite_eq_left (show i < t.singleMembership.bound by omega)]) | |
| constructor | |
| · intro hz | |
| have hzL := level_transitive _ γ _ hv z hz | |
| exact ⟨hzL,hs.mp ((hf z hzL).mpr hz)⟩ | |
| · rintro ⟨hzL,hz⟩ | |
| exact (hf z hzL).mp (hs.mpr hz) | |
| theorem persistent_extension_subset_at_level (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| (I : CountableIdeal) (ρ : I ≃o I) (hρ : Persistent I ρ) | |
| (hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier) | |
| (hIM : I.carrier ⊆ (modelIdeal M hM hT).carrier) : | |
| ∃ σ : modelIdeal M hM hT ≃o modelIdeal M hM hT, | |
| Extends hIM ρ σ ∧ Persistent (modelIdeal M hM hT) σ ∧ | |
| ∃ (t : Construction.{u}) (P : Oracle) (γ : Ordinal.{u}), | |
| P ∈ modelReals M ∧ t.Indexed M ∧ t.Certified (groundReals M) P ∧ γ.toZFSet ∈ M ∧ | |
| (∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧ | |
| automorphismSet σ = definedSubset (level (groundReals M) γ) t.singleMembership | |
| (fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i) := by | |
| obtain ⟨σ,he,hσ,t,P,hP,hi,ht,hval⟩ := persistent_extension_construction M hM hT I ρ hρ hz hIM | |
| obtain ⟨γ,hγ,hin,hdef⟩ := t.subset_at_level M hM hT P hP ht hi | |
| exact ⟨σ,he,hσ,t,P,γ,hP,hi,ht,hγ,hin,hval ▸ hdef⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |