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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 | import OAI.Computability.DegreeRigidity.Constructibility.ContextHierarchyGraph
import OAI.Computability.DegreeRigidity.Syntax.SigmaCollectionClosure
import OAI.Computability.DegreeRigidity.Constructibility.HierarchySyntax
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
theorem hierarchyGraph_sigmaDefinable_collection {M : ZFSet.{u}} (C : Context M) (hRep : SigmaReplacement M) (hColl : SigmaCollection M) (s d r f : ℕ) :
SigmaDefinable M (fun e => HierarchyGraph M (e s) (e d) (e r) (e f)) := by
have hd := (defAllMem (defSubset C 0 (d+1)) d).toSigma C.transitive
have hf := (functionGraph_definable C d r f).toSigma C.transitive
have hv : SigmaDefinable M (fun e => e 1 = definablePower (e 0)) :=
(definablePower_sigmaDefinable_context M C).subst (fun i => if i = 0 then 1 else 0)
have hs := ((stageStep_definable C (s+3) (f+3) (r+3) 2 0).toSigma C.transitive).and hv
have hi := SigmaDefinable.impBounded C (defPairMem C 1 0 (f+2)) hs.existsSet
exact hd.and (hf.and ((hi.allMem_collection C hRep hColl (r+1)).allMem_collection C hRep hColl d))
theorem uniform_hierarchy_certificate_collection (M R d q : ZFSet.{u}) (C : Context M) (hRep : SigmaReplacement M) (hColl : SigmaCollection M) (hs : seed R ∈ M) (hdM : d ∈ M) (hqM : q ∈ M)
(hd : Transitive d) (hq : ∀ c, c ∈ q ↔ c ∈ M ∧ c ⊆ d) :
∃ p : SigmaFormula, ∃ a : ℕ → ZFSet.{u}, (∀ i, a i ∈ M) ∧
∀ x ∈ d, ∀ f ∈ M, p.Realize M (cons f (cons x a)) ↔
∃ r ∈ M, HierarchyGraph M (seed R) (hull d q x) r f := by
have hh : Definable M (fun e => e 1 ∈ q ∧ Transitive (e 1) ∧ e 3 ∈ e 1 ∧
∀ b ∈ q, Transitive b → e 3 ∈ b → e 1 ⊆ b) := by
refine ⟨.isHull 2 6 1 3,cons d (fun _ => q),?_,?_⟩
· intro i; cases i; exact hdM; exact hqM
· intro e
simp [Formula.eval_isHull,mix,cons]
have hg : SigmaDefinable M (fun e => HierarchyGraph M (seed R) (e 1) (e 0) (e 2)) := by
have h := hierarchyGraph_sigmaDefinable_collection C hRep hColl 0 2 1 3
have eq := (equal_param hs 0).toSigma C.transitive
have hx := (eq.and h).existsSet
exact hx.congr (fun e _ => by simp [hs])
have hc := (hh.toSigma C.transitive).and hg
obtain ⟨p,a,ha,hp⟩ := sigma_binary_relation (fun x f =>
∃ c ∈ M, ∃ r ∈ M, (c ∈ q ∧ Transitive c ∧ x ∈ c ∧
∀ b ∈ q, Transitive b → x ∈ b → c ⊆ b) ∧ HierarchyGraph M (seed R) c r f)
hc.existsSet.existsSet
refine ⟨p,a,ha,?_⟩
intro x hx f hf
rw [hp x (C.transitive d hdM x hx) f hf]
constructor
· rintro ⟨c,hc,r,hr,hh,hg⟩
have he := (isHull_iff M d q C.transitive C.separation hdM hqM hd hq hx hc).mp hh
subst c
exact ⟨r,hr,hg⟩
· rintro ⟨r,hr,hg⟩
have hc := hull_mem M d q C.transitive C.separation hdM hqM (C.transitive d hdM x hx)
exact ⟨_,hc,r,hr,(isHull_iff M d q C.transitive C.separation hdM hqM hd hq hx hc).mpr rfl,hg⟩
end TuringRigidity.RelativeConstructible
end OAI
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