File size: 4,118 Bytes
9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | 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
|