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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 56 57 58 59 60 61 62 | import OAI.Computability.DegreeRigidity.Syntax.CheckedDefSuccessor
import OAI.Computability.DegreeRigidity.Constructibility.DomainSyntaxClosure
import OAI.Computability.DegreeRigidity.Constructibility.UniformHierarchyStage
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
def CheckedHierarchyGraph (N w sq s d r f : ZFSet.{u}) : Prop :=
Transitive d ∧ FunctionGraph d r f ∧
∀ x ∈ d, ∀ v ∈ r, ZFSet.pair x v ∈ f →
∃ A ∈ N, StageStep s f r x A ∧ CheckedDefSuccessor N w sq A v
theorem checkedHierarchyGraph_domainDefinable {M : ZFSet.{u}} (C : Context M)
(w sq s d r f : ℕ) :
DomainDefinable M (fun N e => CheckedHierarchyGraph N (e w) (e sq) (e s) (e d) (e r) (e f)) := by
have hd := (defAllMem (defSubset C 0 (d+1)) d).toDomain
have hf := (functionGraph_definable C d r f).toDomain
have hv := (checkedDefSuccessor_domainDefinable C).subst
(fun i => match i with | 0 => 1 | 1 => 0 | 2 => w+3 | _ => sq+3)
have hs := (stageStep_definable C (s+3) (f+3) (r+3) 2 0).toDomain.and hv
have hi := (defPairMem C 1 0 (f+2)).toDomain.imp hs.existsSet
exact hd.and (hf.and ((hi.allMem C (r+1)).allMem C d))
theorem CheckedHierarchyGraph.sound {N s d r f : ZFSet.{u}}
(hf : ∀ p : SentenceForm, family p ∈ N)
(hs : ∀ p : SentenceForm, supportGraph p ∈ N)
(h : CheckedHierarchyGraph N ZFSet.omega (ZFSet.prod ZFSet.omega ZFSet.omega) s d r f) :
HierarchyGraph N s d r f := by
refine ⟨h.1,h.2.1,?_⟩
intro x hx v hv hpair
obtain ⟨A,hA,hstep,hdef⟩ := h.2.2 x hx v hv hpair
exact ⟨A,hA,hstep,hdef.sound hf hs⟩
theorem checkedHierarchyGraph_of_context (N s d r f : ZFSet.{u}) (C : Context N) :
CheckedHierarchyGraph N ZFSet.omega (ZFSet.prod ZFSet.omega ZFSet.omega) s d r f ↔
HierarchyGraph N s d r f := by
constructor
· exact CheckedHierarchyGraph.sound
(family_mem N C.transitive C.pairing C.union C.omega_mem)
(supportGraph_mem N C.transitive C.pairing C.union C.omega_mem)
· intro h
refine ⟨h.1,h.2.1,?_⟩
intro x hx v hv hpair
obtain ⟨A,hA,hstep,hdef⟩ := h.2.2 x hx v hv hpair
exact ⟨A,hA,hstep,(checkedDefSuccessor_of_context N A v C hA).mpr hdef⟩
theorem CheckedHierarchyGraph.stage_exact {N R d r f x A : ZFSet.{u}}
(hf : ∀ p : SentenceForm, family p ∈ N)
(hs : ∀ p : SentenceForm, supportGraph p ∈ N)
(h : CheckedHierarchyGraph N ZFSet.omega (ZFSet.prod ZFSet.omega ZFSet.omega)
(seed R) d r f) (hx : x ∈ d) (hstep : StageStep (seed R) f r x A) :
A = stage R x :=
((h.sound hf hs).stageStep_iff x hx A).mp hstep
end TuringRigidity.RelativeConstructible
end OAI
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