openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /DenseInclusionExtension.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalDenseInclusion | |
| namespace OAI | |
| namespace TuringRigidity.DenseInclusionExtension | |
| open TransitiveNameModel BoundedSetTheory CountableForcing InternalDenseInclusion | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| noncomputable def upperSet (b g : ZFSet.{0}) : ZFSet.{0} := | |
| b.sep (fun q => ∃ p ∈ g, q ⊆ p) | |
| noncomputable def restrictSet (a k : ZFSet.{0}) : ZFSet.{0} := a.sep (fun p => p ∈ k) | |
| theorem upperSet_mem (N b g : ZFSet.{0}) (hN : Transitive N) (hT : SourceT N) | |
| (hb : b ∈ N) (hg : g ∈ N) : upperSet b g ∈ N := by | |
| let e := fun _ : ℕ => g | |
| simpa only [upperSet,Formula.Eval,Formula.eval_subset,cons_zero,cons_succ,e] using | |
| sep_mem N hN hT.separation.finitePrefix.bounded | |
| (.existsMem 1 (.subset 1 0)) e (fun _ => hg) hb | |
| theorem restrictSet_mem (N a k : ZFSet.{0}) (hN : Transitive N) (hT : SourceT N) | |
| (ha : a ∈ N) (hk : k ∈ N) : restrictSet a k ∈ N := by | |
| let e := fun _ : ℕ => k | |
| simpa only [restrictSet,Formula.Eval,cons_zero,cons_succ,e] using | |
| sep_mem N hN hT.separation.finitePrefix.bounded (.member 0 1) e (fun _ => hk) ha | |
| theorem pushFilterSet_eq (a b : ZFSet.{0}) (hab : a ⊆ b) (G : GenericFilter (Conditions a)) : | |
| genericFilterSet b (push a b hab G).carrier = upperSet b (genericFilterSet a G.carrier) := by | |
| apply ZFSet.ext; intro x | |
| constructor | |
| · intro hx | |
| obtain ⟨q,⟨p,hp,hpq⟩,rfl⟩ := (mem_genericFilterSet _ _ x).mp hx | |
| refine ZFSet.mem_sep.mpr ⟨label_mem b q,label a p, | |
| (mem_genericFilterSet _ _ _).mpr ⟨p,hp,rfl⟩,?_⟩ | |
| change label b q ⊆ label b (embed a b hab p) at hpq | |
| rwa [label_embed] at hpq | |
| · intro hx | |
| obtain ⟨hxb,p,hpg,hxp⟩ := ZFSet.mem_sep.mp hx | |
| obtain ⟨q,rfl⟩ := label_surjective b hxb | |
| obtain ⟨p,hp,rfl⟩ := (mem_genericFilterSet _ _ p).mp hpg | |
| exact (mem_genericFilterSet _ _ _).mpr ⟨q,⟨p,hp,by | |
| change label b q ⊆ label b (embed a b hab p); rwa [label_embed]⟩,rfl⟩ | |
| theorem originalFilterSet_eq (a b : ZFSet.{0}) (hab : a ⊆ b) (G : GenericFilter (Conditions a)) : | |
| genericFilterSet a G.carrier = restrictSet a (genericFilterSet b (push a b hab G).carrier) := by | |
| apply ZFSet.ext; intro x | |
| constructor | |
| · intro hx | |
| obtain ⟨p,hp,rfl⟩ := (mem_genericFilterSet _ _ x).mp hx | |
| exact ZFSet.mem_sep.mpr ⟨label_mem a p,(mem_genericFilterSet _ _ _).mpr | |
| ⟨embed a b hab p,(embed_mem_push a b hab G p).mpr hp,label_embed a b hab p⟩⟩ | |
| · intro hx | |
| obtain ⟨hxa,hxk⟩ := ZFSet.mem_sep.mp hx | |
| obtain ⟨p,rfl⟩ := label_surjective a hxa | |
| obtain ⟨q,hq,hqe⟩ := (mem_genericFilterSet _ _ _).mp hxk | |
| have he : q = embed a b hab p := label_injective b (hqe.trans (label_embed a b hab p).symm) | |
| exact (mem_genericFilterSet _ _ _).mpr ⟨p,(embed_mem_push a b hab G p).mp (he ▸ hq),rfl⟩ | |
| theorem extension_eq (M a b : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) | |
| (ha : a ∈ M) (hb : b ∈ M) (hab : a ⊆ b) | |
| (hd : ∀ q ∈ b, ∃ p ∈ a, q ⊆ p) | |
| (G : GenericFilter (Conditions a)) (hG : AtomicForcing.GroundGeneric M G) : | |
| genericExtensionSet M b (push a b hab G).carrier = genericExtensionSet M a G.carrier := by | |
| let K := push a b hab G | |
| have hK := push_ground_generic M a b hM hT ha hb hab hd G hG | |
| obtain ⟨hN,hTN,hMN,hgN⟩ := RegularTreeExtension.extension_properties M a hM hT ha G hG | |
| obtain ⟨hV,hTV,hMV,hkV⟩ := RegularTreeExtension.extension_properties M b hM hT hb K hK | |
| have hkN : genericFilterSet b K.carrier ∈ genericExtensionSet M a G.carrier := by | |
| rw [pushFilterSet_eq] | |
| exact upperSet_mem _ b _ hN hTN (hMN hb) hgN | |
| have hgV : genericFilterSet a G.carrier ∈ genericExtensionSet M b K.carrier := by | |
| rw [originalFilterSet_eq a b hab G] | |
| exact restrictSet_mem _ a _ hV hTV (hMV ha) hkV | |
| apply ZFSet.ext; intro x | |
| exact ⟨fun hx => InternalNameEvaluation.extension_subset M _ hN hTN hMN K.carrier hkN hx, | |
| fun hx => InternalNameEvaluation.extension_subset M _ hV hTV hMV G.carrier hgV hx⟩ | |
| end TuringRigidity.DenseInclusionExtension | |
| end OAI | |