File size: 4,307 Bytes
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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import OAI.Computability.DegreeRigidity.Constructibility.CollectionHierarchySyntax
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
theorem internal_hierarchyGraph_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) (x : ZFSet.{u}) (hx : x ∈ d) :
∃ f ∈ M, HierarchyGraph M (seed R) (hull d q x) (iterUnion 2 f) f := by
let hP := C.pairing
let hU := C.union
let hS := C.separation
let hR := hRep
obtain ⟨p,e,he,hp⟩ := uniform_hierarchy_certificate_collection M R d q C hRep hColl hs hdM hqM hd hq
induction x using ZFSet.inductionOn with
| h x ih =>
have hxM := C.transitive d hdM x hx
have cert (y f : ZFSet.{u}) (hy : y ∈ x) (hf : f ∈ M) :
p.Realize M (cons f (cons y e)) ↔
∃ r ∈ M, HierarchyGraph M (seed R) (hull d q y) r f :=
hp y (hd x hx y hy) f hf
obtain ⟨b,hb,hbdef⟩ := hR p e he x hxM (by
intro y hy
obtain ⟨f,hf,hfg⟩ := ih y hy (hd x hx y hy)
refine ⟨f,hf,(cert y f hy hf).mpr
⟨_,iterUnion_mem M C.transitive hU hf 2,hfg⟩,?_⟩
intro g hg hcg
obtain ⟨r,_,hgg⟩ := (cert y g hy hg).mp hcg
exact hgg.unique hfg)
have hbg (f : ZFSet.{u}) : f ∈ b ↔
∃ y ∈ x, ∃ r ∈ M, HierarchyGraph M (seed R) (hull d q y) r f := by
constructor
· intro hf
have hfM := C.transitive b hb f hf
obtain ⟨y,hy,hc⟩ := (hbdef f hfM).mp hf
exact ⟨y,hy,(cert y f hy hfM).mp hc⟩
· rintro ⟨y,hy,r,hr,hg⟩
obtain ⟨f',hf',hg'⟩ := ih y hy (hd x hx y hy)
have heq := hg.unique hg'
have hfM : f ∈ M := heq ▸ hf'
exact (hbdef f hfM).mpr ⟨y,hy,(cert y f hy hfM).mpr ⟨r,hr,hg⟩⟩
let g := ZFSet.sUnion b
have hgM : g ∈ M := union_mem M C.transitive hU hb
have hg (z : ZFSet.{u}) : z ∈ g ↔
∃ y ∈ x, ∃ w ∈ hull d q y, z = ZFSet.pair w (definablePower (stage R w)) := by
constructor
· intro hz
obtain ⟨f,hf,hzf⟩ := ZFSet.mem_sUnion.mp hz
obtain ⟨y,hy,r,_,hfg⟩ := (hbg f).mp hf
exact ⟨y,hy,(hfg.mem_iff z).mp hzf⟩
· rintro ⟨y,hy,w,hw,hz⟩
obtain ⟨f,hf,hfg⟩ := ih y hy (hd x hx y hy)
exact ZFSet.mem_sUnion.mpr ⟨f,(hbg f).mpr
⟨y,hy,_,iterUnion_mem M C.transitive hU hf 2,hfg⟩,(hfg.mem_iff z).mpr ⟨w,hw,hz⟩⟩
have hvalue : stageImage (seed R) x g = stage R x := by
apply ZFSet.ext; intro z
rw [mem_stageImage,mem_stage]
apply or_congr_right
constructor
· rintro ⟨y,hy,w,hw,hz⟩
obtain ⟨a,_,v,_,hp⟩ := (hg _).mp hw
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp hp
exact ⟨y,hy,hz⟩
· rintro ⟨y,hy,hz⟩
exact ⟨y,hy,_,(hg _).mpr
⟨y,hy,y,self_mem_hull d q (hd x hx y hy),rfl⟩,hz⟩
have hvM : definablePower (stage R x) ∈ M := definablePower_mem_of_context M _ C
(hvalue ▸ stageImage_mem_context M C hs hxM hgM)
let f := g ∪ ({ZFSet.pair x (definablePower (stage R x))} : ZFSet.{u})
have hfM : f ∈ M := binary_union_mem M C.transitive hP hU hgM
(singleton_mem M C.transitive hP (orderedPair_mem M C.transitive hP hxM hvM))
refine ⟨f,hfM,hierarchyGraph_of_mem_iff_context M R _ f C hs
(hull_mem M d q C.transitive hS hdM hqM hxM) hfM (hull_transitive d q x hd) ?_⟩
intro z
change z ∈ g ∪ ({ZFSet.pair x (definablePower (stage R x))} : ZFSet.{u}) ↔ _
rw [ZFSet.mem_union,ZFSet.mem_singleton,hg]
constructor
· rintro (⟨y,hy,w,hw,hz⟩|hz)
· exact ⟨w,(hull_unfold M d q C.transitive hP hU hS hR hdM hqM hd hq hx w).mpr
(Or.inr ⟨y,hy,hw⟩),hz⟩
· exact ⟨x,self_mem_hull d q hx,hz⟩
· rintro ⟨w,hw,hz⟩
rcases (hull_unfold M d q C.transitive hP hU hS hR hdM hqM hd hq hx w).mp hw with rfl|⟨y,hy,hw⟩
· exact Or.inr hz
· exact Or.inl ⟨y,hy,w,hw,hz⟩
end TuringRigidity.RelativeConstructible
end OAI
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