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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.IntersectionName
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory CountableForcing
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)] [Top (Conditions c)]
noncomputable def powerName (a : Name (Conditions c)) (W : ZFSet.{u}) : Name (Conditions c) :=
.mk (Conditions W) (fun s => a.restrict (label c) (label W s)) (fun _ => ⊤)
theorem encode_powerName (a : Name (Conditions c)) (W : ZFSet.{u})
(hW : ∀ B ∈ W, B ⊆ a.support (label c)) :
(powerName a W).encode (label c) = ZFSet.prod W ({label c ⊤} : ZFSet.{u}) := by
apply ZFSet.ext
intro z
change z ∈ ZFSet.range _ ↔ _
rw [ZFSet.mem_range]
constructor
· rintro ⟨s,hs⟩
rw [Name.encode_restrict _ _ _ (hW _ (label_mem W s))] at hs
exact ZFSet.mem_prod.mpr ⟨_,label_mem W s,_,ZFSet.mem_singleton.mpr rfl,hs.symm⟩
· intro hz
obtain ⟨B,hB,t,ht,hz⟩ := ZFSet.mem_prod.mp hz
obtain rfl := ZFSet.mem_singleton.mp ht
obtain ⟨s,rfl⟩ := label_surjective W hB
exact ⟨s,by rw [Name.encode_restrict _ _ _ (hW _ hB)]; exact hz.symm⟩
theorem mem_val_powerName (a : Name (Conditions c)) (W : ZFSet.{u})
(G : Set (Conditions c)) (hG : ⊤ ∈ G) (x : ZFSet.{u}) :
x ∈ (powerName a W).val G ↔ ∃ B ∈ W, (a.restrict (label c) B).val G = x := by
rw [powerName,Name.mem_val]
constructor
· rintro ⟨s,_,hs⟩
exact ⟨_,label_mem W s,hs⟩
· rintro ⟨B,hB,hx⟩
obtain ⟨s,rfl⟩ := label_surjective W hB
exact ⟨s,hG,hx⟩
theorem extension_powerSet (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
(hR : SigmaReplacement M) (hI : Infinity M) (hc : c ∈ M)
{o : ZFSet.{u}} (hoM : o ∈ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(G : GenericFilter (Conditions c)) (hG : GroundGeneric M G) (htop : ⊤ ∈ G.carrier) :
PowerSet (genericExtensionSet M c G.carrier) := by
intro x hx
obtain ⟨a,ha,rfl⟩ := (mem_extensionSet M c G.carrier x).mp hx
have hsupport := support_mem M hM hP hU hPow hS hc hoM ho a ha
obtain ⟨W,hW,hWdef⟩ := internal_power M hM hPow hsupport
have hWsub : ∀ B ∈ W, B ⊆ a.support (label c) := fun B hB => ((hWdef B).mp hB).2
have hpower : (powerName a W).encode (label c) ∈ M := by
rw [encode_powerName a W hWsub]
exact product_mem M hM hP hU hPow hS hW
(singleton_mem M hM hP (hM c hc _ (label_mem c ⊤)))
refine ⟨(powerName a W).val G.carrier,(mem_extensionSet M c G.carrier _).mpr
⟨_,hpower,rfl⟩,?_⟩
intro z hz
constructor
· intro hzPower y _ hyz
obtain ⟨B,hB,hBz⟩ := (mem_val_powerName a W G.carrier htop z).mp hzPower
rw [←hBz] at hyz
exact a.val_restrict_subset (label c) B G hyz
· intro hsub
obtain ⟨b,hb,rfl⟩ := (mem_extensionSet M c G.carrier z).mp hz
have hba : b.val G.carrier ⊆ a.val G.carrier := fun y hy => hsub y
(genericExtensionSet_transitive M c hM G.carrier _
((mem_extensionSet M c G.carrier _).mpr ⟨b,hb,rfl⟩) y hy) hy
obtain ⟨B,hBM,hBS,hBval⟩ := internal_intersection_name M hM hP hU hPow hS hR hI hc hoM ho G hG a b ha hb
apply (mem_val_powerName a W G.carrier htop _).mpr
refine ⟨B,(hWdef B).mpr ⟨hBM,hBS⟩,hBval.trans ?_⟩
apply ZFSet.ext
intro y
rw [ZFSet.mem_inter]
exact ⟨And.right,fun hy => ⟨hba hy,hy⟩⟩
end TuringRigidity.AtomicForcing
end OAI
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