File size: 4,047 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 | import OAI.Computability.DegreeRigidity.SetModels.ChoiceFromOrdinalCover
import OAI.Computability.DegreeRigidity.SetModels.NameEvaluationGraph
import OAI.Computability.DegreeRigidity.SetModels.ExtensionSigmaReplacement
import OAI.Computability.DegreeRigidity.SetModels.ExtensionPowerSet
namespace OAI
namespace TuringRigidity.BoundedForcing
open RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing CountableForcing
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)] [Top (Conditions c)]
theorem extension_choice (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : SigmaSeparation M) (hR : SigmaReplacement M) (hI : Infinity M) (hAC : Choice 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) :
Choice (genericExtensionSet M c G.carrier) := by
classical
let E := genericExtensionSet M c G.carrier
have hET : Transitive E := genericExtensionSet_transitive M c hM G.carrier
have hEP : Pairing E := extension_pairing M c hM hP hc G.carrier htop
have hEU : BoundedSetTheory.Union E := extension_union M c o hM hP hU hPow hS.bounded hc hoM ho G
have hEPow : PowerSet E := extension_powerSet M hM hP hU hPow hS.bounded hR hI hc hoM ho G hG htop
have hES : Separation E := (extension_sigmaSeparation M hM hP hU hPow hS hR hI hc hoM ho G hG).bounded
have hground : M ⊆ E := ground_inclusion_set M c hM hP hU hPow hS.bounded hR hI hc G.carrier htop
intro X hX hnonempty
obtain ⟨a,ha,rfl⟩ := (mem_extensionSet M c G.carrier X).mp hX
obtain ⟨d,hdM,hd,had⟩ := internal_transitive_container M hM hP hU hS.bounded hR hI ha
let V := ZFSet.sep (fun x => (NameValidity.Code.valid 1 0).Realize M (cons x (fun _ => c))) d
have hV : V ∈ M := sigma_sep_mem M hM hS (NameValidity.Code.valid 1 0) (fun _ => c) (fun _ => hc) hdM
have valid (x : ZFSet.{u}) (hx : x ∈ M) :
(NameValidity.Code.valid 1 0).Realize M (cons x (fun _ => c)) ↔
∃ b : Name (Conditions c), b.encode (label c) = x :=
NameValidity.realize_valid M hM hP hU hPow hS.bounded hR hI _
(by intro i; cases i <;> assumption) 1 0
have hvalid (x : ZFSet.{u}) (hx : x ∈ V) : ∃ b : Name (Conditions c), b.encode (label c) = x :=
(valid x (hM V hV x hx)).mp (ZFSet.mem_sep.mp hx).2
have code_mem (b : Name (Conditions c)) (hb : b.encode (label c) ∈ d) : b.encode (label c) ∈ V :=
ZFSet.mem_sep.mpr ⟨hb,(valid _ (hM d hdM _ hb)).mpr ⟨b,rfl⟩⟩
obtain ⟨δ,hδM,hδ,F,hFM,hF,hsurj,_⟩ :=
InternalWellOrder.internal_ordinal_enumeration M hM hP hU hPow hS hR hI hAC hV
obtain ⟨N,hN,hNdef⟩ := internal_name_evaluation_graph M hM hP hU hPow hS.bounded hR hI hc
hV hvalid G.carrier htop
have hNE : N.val G.carrier ∈ E := (mem_extensionSet M c G.carrier _).mpr ⟨N,hN,rfl⟩
have hNfun : FiniteTerm.Functional (N.val G.carrier) := by
intro v x y hx hy
obtain ⟨v',_,b,hb,he⟩ := (hNdef _).mp hx
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
obtain ⟨v'',_,b',hb',he'⟩ := (hNdef _).mp hy
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he'
exact Name.val_eq_of_encode_eq (label c) (label_injective c) G.carrier _ _ (hb.trans hb'.symm)
apply InternalWellOrder.internal_choice_of_ordinal_cover E hET hEP hEU hEPow hES
hX (hground hδM) (hground hV) (hground hFM) hNE hδ hF hNfun
intro x hx
obtain ⟨y,hy⟩ := hnonempty x hx
obtain ⟨b,hbd,hbx⟩ := name_of_value_member d hd a had G.carrier x hx
have hyb : y ∈ b.val G.carrier := hbx.symm ▸ hy
obtain ⟨s,hsd,hsy⟩ := name_of_value_member d hd b hbd G.carrier y hyb
have hsV := code_mem s hsd
obtain ⟨i,hi,hif⟩ := hsurj _ hsV
refine ⟨i,hi,s.encode (label c),hsV,y,hy,hif,(hNdef _).mpr ?_⟩
exact ⟨_,hsV,s,rfl,congrArg (ZFSet.pair (s.encode (label c))) hsy.symm⟩
end TuringRigidity.BoundedForcing
end OAI
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