openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /CountableGroundGeneric.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalAtomicTruth | |
| namespace OAI | |
| namespace TuringRigidity.AtomicForcing | |
| open TransitiveNameModel BoundedSetTheory CountableForcing | |
| universe u | |
| theorem countable_ground_generic (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hne : ∃ x, x ∈ M) {c : ZFSet.{u}} [Preorder (Conditions c)] (p : Conditions c) : | |
| ∃ G : GenericFilter (Conditions c), p ∈ G.carrier ∧ GroundGeneric M G := by | |
| classical | |
| obtain ⟨x,hx⟩ := hne | |
| obtain ⟨m,_⟩ := label_surjective M hx | |
| let _ : Nonempty (Conditions M) := ⟨m⟩ | |
| obtain ⟨e,he⟩ := exists_surjective_nat (Conditions M) | |
| let D : ℕ → Set (Conditions c) := fun n => | |
| if Dense {q : Conditions c | label c q ∈ label M (e n)} then | |
| {q | label c q ∈ label M (e n)} else Set.univ | |
| have hD : ∀ n, Dense (D n) := by | |
| intro n | |
| dsimp only [D] | |
| split | |
| · assumption | |
| · exact fun q => ⟨q,le_rfl,Set.mem_univ _⟩ | |
| obtain ⟨G,hp,hG⟩ := exists_generic D hD p | |
| refine ⟨G,hp,?_⟩ | |
| intro E hEM hE | |
| obtain ⟨m,hm⟩ := label_surjective M hEM | |
| obtain ⟨n,rfl⟩ := he m | |
| have hd : Dense {q : Conditions c | label c q ∈ label M (e n)} := by | |
| rw [hm]; exact hE | |
| obtain ⟨q,hq,hqn⟩ := hG n | |
| refine ⟨q,hq,?_⟩ | |
| simpa only [D,ite_eq_left hd,Set.mem_ofPred_eq,hm] using hqn | |
| end TuringRigidity.AtomicForcing | |
| end OAI | |