openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /ElementaryModelIdeal.lean
| import OAI.Computability.DegreeRigidity.SetModels.ElementaryDegreeHull | |
| namespace OAI | |
| namespace TuringRigidity.ElementaryModel | |
| open BoundedSetTheory TransitiveNameModel RelationCollapse SetDegreeDecoding PersistentRestrictions OracleJump | |
| universe u | |
| theorem model_reals_countable (M : ZFSet.{u}) [Countable (Conditions M)] : (modelReals M).Countable := by | |
| have hc : (M : Set ZFSet.{u}).Countable := by | |
| apply (Set.countable_range (label M)).mono | |
| intro x hx; exact label_surjective M hx | |
| apply (hc.image decodeReal).mono | |
| intro A hA | |
| exact ⟨realCode A,hA,decodeReal_realCode A⟩ | |
| noncomputable def modelIdeal (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M) : | |
| CountableIdeal where | |
| carrier := {b | ∃ A ∈ modelReals M, degree A = b} | |
| nonempty := ⟨degree FixedArithmetic.zero,FixedArithmetic.zero,sourceT_zero_real M hM hT,rfl⟩ | |
| countable := (model_reals_countable M).image degree | |
| lower := by | |
| intro a b hab hb | |
| obtain ⟨A,rfl⟩ := degree_surjective a | |
| obtain ⟨B,hB,rfl⟩ := hb | |
| exact ⟨A,sourceT_real_lower M hM hT hB hab,rfl⟩ | |
| join_mem := by | |
| rintro a b ⟨A,hA,rfl⟩ ⟨B,hB,rfl⟩ | |
| exact ⟨TuringRigidity.join A B,sourceT_real_join M hM hT hA hB,rfl⟩ | |
| theorem modelIdeal_mem (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M) | |
| (b : Degree) : b ∈ (modelIdeal M hM hT).carrier ↔ ∃ A ∈ modelReals M, degree A = b := Iff.rfl | |
| theorem modelIdeal_jump (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M) : | |
| IdealLocality.JumpClosed (PersistentLocality.ideal (modelIdeal M hM hT)) := by | |
| rintro b ⟨A,hA,rfl⟩ | |
| exact ⟨jump A,sourceT_real_jump M hM hT hA,rfl⟩ | |
| theorem hull_degree_iff (a : ℕ → ZFSet.{u}) | |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) | |
| (hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a) (b : Degree) : | |
| b ∈ (modelIdeal (collapsed (hullSet a)) (collapsed_transitive _) (collapsed_hull_sourceT a)).carrier ↔ | |
| degreeSet b ∈ hullSet a := by | |
| constructor | |
| · rintro ⟨A,hA,rfl⟩ | |
| exact degreeSet_mem_hull a hω (hn 0) hR hQ ((collapsed_hull_reals a hn hω A).mp hA) | |
| · intro hb | |
| obtain ⟨A,hA,hAb⟩ := degree_representative_in_hull a b hb | |
| exact ⟨A,(collapsed_hull_reals a hn hω A).mpr hA,hAb⟩ | |
| theorem hull_ideal_invariant (a : ℕ → ZFSet.{u}) | |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) | |
| (hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a) | |
| (π : Degree ≃o Degree) (hπ : globalGraph π ∈ hullSet a) : | |
| let I := modelIdeal (collapsed (hullSet a)) (collapsed_transitive _) (collapsed_hull_sourceT a) | |
| ∀ b ∈ I.carrier, π b ∈ I.carrier ∧ π.symm b ∈ I.carrier := by | |
| intro I b hb | |
| have hd := (hull_degree_iff a hn hω hR hQ b).mp hb | |
| have hi := image_degreeSet_mem_hull a π hπ b hd | |
| exact ⟨(hull_degree_iff a hn hω hR hQ _).mpr hi.1,(hull_degree_iff a hn hω hR hQ _).mpr hi.2⟩ | |
| end TuringRigidity.ElementaryModel | |
| end OAI | |