openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /CommonGenericModels.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalDenseFamily | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseExistence | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory.InternalGeneric | |
| open TransitiveNameModel CountableForcing | |
| universe u | |
| instance extension_countable (M c : ZFSet.{u}) [Countable (Conditions M)] (G : Set (Conditions c)) : | |
| Countable (Conditions (genericExtensionSet M c G)) := by | |
| have hM : (M : Set ZFSet.{u}).Countable := by | |
| apply (Set.countable_range (label M)).mono | |
| intro x hx | |
| exact label_surjective M hx | |
| have hE : (genericExtensionSet M c G : Set ZFSet.{u}).Countable := by | |
| rw [genericExtensionSet_coe] | |
| exact genericExtension_countable M c hM G | |
| let : Countable (genericExtensionSet M c G : Set ZFSet.{u}) := hE.to_subtype | |
| exact Function.Injective.countable (f := fun p : Conditions (genericExtensionSet M c G) => | |
| (⟨label _ p,label_mem _ p⟩ : (genericExtensionSet M c G : Set ZFSet.{u}))) | |
| (fun x y h => label_injective _ (congrArg (fun z : (genericExtensionSet M c G : Set ZFSet.{u}) => z.val) h)) | |
| theorem counting_model_with_generic (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) {c o : ZFSet.{u}} [Preorder (Conditions c)] | |
| (hc : c ∈ M) (hoM : o ∈ M) | |
| (ho : ∀ p q : Conditions c, ZFSet.pair (label c p) (label c q) ∈ o ↔ p ≤ q) | |
| (p₀ : Conditions c) : | |
| ∃ N : ZFSet.{u}, M ⊆ N ∧ Transitive N ∧ SourceT N ∧ Countable (Conditions N) ∧ | |
| ∃ G : GenericFilter (Conditions c), p₀ ∈ G.carrier ∧ | |
| genericFilterSet c G.carrier ∈ N ∧ AtomicForcing.GroundGeneric M G := by | |
| obtain ⟨D,hDM,hD,hcD⟩ := dense_family M hM hT hc hoM ho | |
| obtain ⟨d,hd,_,_,H,_,hMN,hN,hTN,hcount⟩ := | |
| InternalCollapse.exists_internal_counting_extension M hM hT hDM ⟨c,hcD⟩ | |
| let : Preorder (Conditions d) := InternalCollapse.collapsePreorder d | |
| let N := genericExtensionSet M d H.carrier | |
| obtain ⟨G,hp,hGN,hG⟩ := internal_ground_generic M N hM hN hT hTN hMN hc hoM hDM ho hD hcount p₀ | |
| exact ⟨N,hMN,hN,hTN,inferInstance,G,hp,hGN,hG⟩ | |
| theorem common_generic_model (M : ZFSet.{u}) [Countable (Conditions M)] | |
| (hM : Transitive M) (hT : SourceT M) | |
| {c o d v : ZFSet.{u}} [Preorder (Conditions c)] [Preorder (Conditions d)] | |
| (hc : c ∈ M) (hoM : o ∈ M) | |
| (ho : ∀ p q : Conditions c, ZFSet.pair (label c p) (label c q) ∈ o ↔ p ≤ q) | |
| (hd : d ∈ M) (hvM : v ∈ M) | |
| (hv : ∀ p q : Conditions d, ZFSet.pair (label d p) (label d q) ∈ v ↔ p ≤ q) | |
| (p₀ : Conditions c) (q₀ : Conditions d) : | |
| ∃ N : ZFSet.{u}, M ⊆ N ∧ Transitive N ∧ SourceT N ∧ Countable (Conditions N) ∧ | |
| ∃ G : GenericFilter (Conditions c), ∃ H : GenericFilter (Conditions d), | |
| p₀ ∈ G.carrier ∧ q₀ ∈ H.carrier ∧ genericFilterSet c G.carrier ∈ N ∧ | |
| genericFilterSet d H.carrier ∈ N ∧ AtomicForcing.GroundGeneric M G ∧ | |
| AtomicForcing.GroundGeneric M H := by | |
| obtain ⟨N,hMN,hN,hTN,hcount,G,hp,hGN,hG⟩ := counting_model_with_generic M hM hT hc hoM ho p₀ | |
| let : Countable (Conditions N) := hcount | |
| obtain ⟨K,hNK,hK,hTK,hcountK,H,hq,hHK,hH⟩ := | |
| counting_model_with_generic N hN hTN (hMN hd) (hMN hvM) hv q₀ | |
| refine ⟨K,fun _ hx => hNK (hMN hx),hK,hTK,hcountK,G,H,hp,hq,hNK hGN,hHK,hG,?_⟩ | |
| intro E hEM hE | |
| exact hH E (hMN hEM) hE | |
| end TuringRigidity.BoundedSetTheory.InternalGeneric | |
| end OAI | |