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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.LevelInlinedDefinition
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SetDegreeDecoding PersistentRestrictions
universe u
theorem Construction.subset_at_level (t : Construction.{u}) (M : ZFSet.{u})
(hM : Transitive M) (hT : SourceT M) (P : Oracle)
(hP : P ∈ modelReals M) (ht : t.Certified (groundReals M) P) (hi : t.Indexed M) :
∃ γ : Ordinal.{u}, γ.toZFSet ∈ M ∧
(∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧
t.value (groundReals M) P = definedSubset (level (groundReals M) γ) t.singleMembership
(fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i) := by
obtain ⟨γ,hγ,hv,he,hf⟩ := t.inlined_at_level M hM hT P hP ht hi
refine ⟨γ,hγ,he,?_⟩
apply ZFSet.ext; intro z
rw [mem_definedSubset]
have hs := t.singleMembership.finite_support (level (groundReals M) γ : Set ZFSet)
(cons z (t.singleInputs (groundReals M) P))
(cons z (tupleEnv (fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i)))
(by
intro i hin
cases i with
| zero => rfl
| succ i =>
simp only [cons_succ,tupleEnv,dite_eq_left (show i < t.singleMembership.bound by omega)])
constructor
· intro hz
have hzL := level_transitive _ γ _ hv z hz
exact ⟨hzL,hs.mp ((hf z hzL).mpr hz)⟩
· rintro ⟨hzL,hz⟩
exact (hf z hzL).mp (hs.mpr hz)
theorem persistent_extension_subset_at_level (M : ZFSet.{u}) [Countable (Conditions M)]
(hM : Transitive M) (hT : SourceT M)
(I : CountableIdeal) (ρ : I ≃o I) (hρ : Persistent I ρ)
(hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier)
(hIM : I.carrier ⊆ (modelIdeal M hM hT).carrier) :
∃ σ : modelIdeal M hM hT ≃o modelIdeal M hM hT,
Extends hIM ρ σ ∧ Persistent (modelIdeal M hM hT) σ ∧
∃ (t : Construction.{u}) (P : Oracle) (γ : Ordinal.{u}),
P ∈ modelReals M ∧ t.Indexed M ∧ t.Certified (groundReals M) P ∧ γ.toZFSet ∈ M ∧
(∀ i, t.singleInputs (groundReals M) P i ∈ level (groundReals M) γ) ∧
automorphismSet σ = definedSubset (level (groundReals M) γ) t.singleMembership
(fun i : Fin t.singleMembership.bound => t.singleInputs (groundReals M) P i) := by
obtain ⟨σ,he,hσ,t,P,hP,hi,ht,hval⟩ := persistent_extension_construction M hM hT I ρ hρ hz hIM
obtain ⟨γ,hγ,hin,hdef⟩ := t.subset_at_level M hM hT P hP ht hi
exact ⟨σ,he,hσ,t,P,γ,hP,hi,ht,hγ,hin,hval ▸ hdef⟩
end TuringRigidity.RelativeConstructible
end OAI
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