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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 | 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
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