| import OAI.Computability.DegreeRigidity.Syntax.VariableCertificateBounds |
| import OAI.Computability.DegreeRigidity.Constructibility.DomainParameterizedSyntax |
| import OAI.Computability.DegreeRigidity.Constructibility.ContextDefinablePower |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.RelativeConstructible |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding |
| open BoundedDefinability SetModelFunctions |
| universe u |
|
|
| def CheckedDefSuccessor (N w s A D : ZFSet.{u}) : Prop := |
| ∃ Q ∈ N, ∃ B ∈ N, (PowerBound N w Q ∧ PowerBound N s B) ∧ |
| ∃ G ∈ N, ∃ T ∈ N, ∃ H ∈ N, ∃ Z ∈ N, DefSystem Q B A G T H Z D |
|
|
| theorem checkedDefSuccessor_domainDefinable {M : ZFSet.{u}} (C : Context M) : |
| DomainDefinable M (fun N e => CheckedDefSuccessor N (e 2) (e 3) (e 1) (e 0)) := by |
| have hs := (defSystem_allBounds C 5 4 7 3 2 1 0 6).toDomain.existsSet.existsSet.existsSet.existsSet |
| have hb := (powerBound_domainDefinable C 4 1).and (powerBound_domainDefinable C 5 0) |
| exact (hb.and hs).existsSet.existsSet |
|
|
| theorem CheckedDefSuccessor.sound {N A D : ZFSet.{u}} |
| (hf : ∀ p : SentenceForm, family p ∈ N) |
| (hs : ∀ p : SentenceForm, supportGraph p ∈ N) |
| (h : CheckedDefSuccessor N ZFSet.omega (ZFSet.prod ZFSet.omega ZFSet.omega) A D) : |
| D = definablePower A := by |
| obtain ⟨Q,_,B,_,hb,G,_,T,_,H,_,Z,_,hd⟩ := h |
| exact hd.exact (fun p => (hb.1 _).mpr ⟨hf p,family_subset_omega p⟩) |
| (fun p => (hb.2 _).mpr ⟨hs p,supportGraph_subset p⟩) |
|
|
| theorem checkedDefSuccessor_of_context (M A D : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : |
| CheckedDefSuccessor M ZFSet.omega (ZFSet.prod ZFSet.omega ZFSet.omega) A D ↔ |
| D = definablePower A := by |
| constructor |
| · exact CheckedDefSuccessor.sound |
| (family_mem M C.transitive C.pairing C.union C.omega_mem) |
| (supportGraph_mem M C.transitive C.pairing C.union C.omega_mem) |
| · rintro rfl |
| obtain ⟨Q,hQ,hq,hf⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem |
| obtain ⟨B,hB,hb⟩ := internal_power M C.transitive C.power (C.prod_mem C.omega_mem C.omega_mem) |
| have hs (p : SentenceForm) : supportGraph p ∈ B := (hb _).mpr |
| ⟨supportGraph_mem M C.transitive C.pairing C.union C.omega_mem p,supportGraph_subset p⟩ |
| exact ⟨Q,hQ,B,hB,⟨hq,hb⟩,internal_defSystem_of_context M C Q B A hA hf hs⟩ |
|
|
| end TuringRigidity.RelativeConstructible |
|
|
| end OAI |
|
|
