File size: 3,100 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 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryDegreeHull
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse SetDegreeDecoding PersistentRestrictions OracleJump
universe u
theorem model_reals_countable (M : ZFSet.{u}) [Countable (Conditions M)] : (modelReals M).Countable := by
have hc : (M : Set ZFSet.{u}).Countable := by
apply (Set.countable_range (label M)).mono
intro x hx; exact label_surjective M hx
apply (hc.image decodeReal).mono
intro A hA
exact ⟨realCode A,hA,decodeReal_realCode A⟩
noncomputable def modelIdeal (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M) :
CountableIdeal where
carrier := {b | ∃ A ∈ modelReals M, degree A = b}
nonempty := ⟨degree FixedArithmetic.zero,FixedArithmetic.zero,sourceT_zero_real M hM hT,rfl⟩
countable := (model_reals_countable M).image degree
lower := by
intro a b hab hb
obtain ⟨A,rfl⟩ := degree_surjective a
obtain ⟨B,hB,rfl⟩ := hb
exact ⟨A,sourceT_real_lower M hM hT hB hab,rfl⟩
join_mem := by
rintro a b ⟨A,hA,rfl⟩ ⟨B,hB,rfl⟩
exact ⟨TuringRigidity.join A B,sourceT_real_join M hM hT hA hB,rfl⟩
theorem modelIdeal_mem (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M)
(b : Degree) : b ∈ (modelIdeal M hM hT).carrier ↔ ∃ A ∈ modelReals M, degree A = b := Iff.rfl
theorem modelIdeal_jump (M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M) :
IdealLocality.JumpClosed (PersistentLocality.ideal (modelIdeal M hM hT)) := by
rintro b ⟨A,hA,rfl⟩
exact ⟨jump A,sourceT_real_jump M hM hT hA,rfl⟩
theorem hull_degree_iff (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a)
(hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a) (b : Degree) :
b ∈ (modelIdeal (collapsed (hullSet a)) (collapsed_transitive _) (collapsed_hull_sourceT a)).carrier ↔
degreeSet b ∈ hullSet a := by
constructor
· rintro ⟨A,hA,rfl⟩
exact degreeSet_mem_hull a hω (hn 0) hR hQ ((collapsed_hull_reals a hn hω A).mp hA)
· intro hb
obtain ⟨A,hA,hAb⟩ := degree_representative_in_hull a b hb
exact ⟨A,(collapsed_hull_reals a hn hω A).mpr hA,hAb⟩
theorem hull_ideal_invariant (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a)
(hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a)
(π : Degree ≃o Degree) (hπ : globalGraph π ∈ hullSet a) :
let I := modelIdeal (collapsed (hullSet a)) (collapsed_transitive _) (collapsed_hull_sourceT a)
∀ b ∈ I.carrier, π b ∈ I.carrier ∧ π.symm b ∈ I.carrier := by
intro I b hb
have hd := (hull_degree_iff a hn hω hR hQ b).mp hb
have hi := image_degreeSet_mem_hull a π hπ b hd
exact ⟨(hull_degree_iff a hn hω hR hQ _).mpr hi.1,(hull_degree_iff a hn hω hR hQ _).mpr hi.2⟩
end TuringRigidity.ElementaryModel
end OAI
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