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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 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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Computability.DegreeRigidity.Constructibility.UniformDefSuccessor
import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleStages
import OAI.Computability.DegreeRigidity.Syntax.CheckGraphConstruction
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
def StageStep (s f r x A : ZFSet.{u}) : Prop :=
s ⊆ A ∧
(∀ z ∈ A, z ∈ s ∨ ∃ y ∈ x, ∃ w ∈ r, ZFSet.pair y w ∈ f ∧ z ∈ w) ∧
∀ y ∈ x, ∀ w ∈ r, ZFSet.pair y w ∈ f → w ⊆ A
def HierarchyGraph (M s d r f : ZFSet.{u}) : Prop :=
Transitive d ∧ FunctionGraph d r f ∧
∀ x ∈ d, ∀ v ∈ r, ZFSet.pair x v ∈ f →
∃ A ∈ M, StageStep s f r x A ∧ v = definablePower A
theorem HierarchyGraph.correct {M R d r f : ZFSet.{u}}
(h : HierarchyGraph M (seed R) d r f) (x : ZFSet.{u}) (hx : x ∈ d)
(v : ZFSet.{u}) (hv : v ∈ r) (hfv : ZFSet.pair x v ∈ f) :
v = definablePower (stage R x) := by
induction x using ZFSet.inductionOn generalizing v with
| h x ih =>
obtain ⟨A,_,hs,rfl⟩ := h.2.2 x hx v hv hfv
congr 1
apply ZFSet.ext; intro z
rw [mem_stage]
constructor
· intro hz
rcases hs.2.1 z hz with hz|⟨y,hy,w,hw,hfw,hz⟩
· exact Or.inl hz
· rw [ih y hy (h.1 x hx y hy) w hw hfw] at hz
exact Or.inr ⟨y,hy,hz⟩
· rintro (hz|⟨y,hy,hz⟩)
· exact hs.1 hz
· obtain ⟨w,hw,hfw,_⟩ := h.2.1.2 y (h.1 x hx y hy)
rw [← ih y hy (h.1 x hx y hy) w hw hfw] at hz
exact hs.2.2 y hy w hw hfw hz
theorem HierarchyGraph.mem_iff {M R d r f : ZFSet.{u}}
(h : HierarchyGraph M (seed R) d r f) (z : ZFSet.{u}) :
z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x)) := by
constructor
· intro hz
obtain ⟨x,hx,v,hv,rfl⟩ := h.2.1.1 z hz
exact ⟨x,hx,by rw [h.correct x hx v hv hz]⟩
· rintro ⟨x,hx,rfl⟩
obtain ⟨v,hv,hfv,_⟩ := h.2.1.2 x hx
rw [← h.correct x hx v hv hfv]
exact hfv
theorem HierarchyGraph.unique {M N R d r t f g : ZFSet.{u}}
(hf : HierarchyGraph M (seed R) d r f) (hg : HierarchyGraph N (seed R) d t g) : f = g := by
apply ZFSet.ext; intro z
exact (hf.mem_iff z).trans (hg.mem_iff z).symm
noncomputable def stageImage (s x f : ZFSet.{u}) : ZFSet.{u} :=
s ∪ (ZFSet.sUnion (iterUnion 2 f)).sep
(fun z => ∃ y ∈ x, ∃ w ∈ iterUnion 2 f, ZFSet.pair y w ∈ f ∧ z ∈ w)
theorem mem_stageImage (s x f z : ZFSet.{u}) :
z ∈ stageImage s x f ↔ z ∈ s ∨ ∃ y ∈ x, ∃ w, ZFSet.pair y w ∈ f ∧ z ∈ w := by
rw [stageImage,ZFSet.mem_union,ZFSet.mem_sep]
apply or_congr_right
constructor
· rintro ⟨_,y,hy,w,_,hf,hz⟩; exact ⟨y,hy,w,hf,hz⟩
· rintro ⟨y,hy,w,hf,hz⟩
have hw := second_mem_doubleUnion hf
exact ⟨ZFSet.mem_sUnion.mpr ⟨w,hw,hz⟩,y,hy,w,hw,hf,hz⟩
theorem stageImage_mem (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
{s x f : ZFSet.{u}} (hs : s ∈ M) (hx : x ∈ M) (hf : f ∈ M) : stageImage s x f ∈ M := by
let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩
have hr := iterUnion_mem M hM hT.union hf 2
have hd := (((defPairMem C 1 0 4).and (member_definable C 2 0)).existsMem 4).existsMem 1
let e := cons x (cons f (fun _ => iterUnion 2 f))
have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hx; exact hf; exact hr
exact binary_union_mem M hM hT.pairing hT.union hs
(hd.sep_mem C e he (union_mem M hM hT.union hr))
theorem hierarchyGraph_of_mem_iff (M R d f : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(hs : seed R ∈ M) (hdM : d ∈ M) (hfM : f ∈ M) (hd : Transitive d)
(hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x))) :
HierarchyGraph M (seed R) d (iterUnion 2 f) f := by
have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = definablePower (stage R x) := by
rw [hf]
constructor
· rintro ⟨y,hy,he⟩
obtain ⟨rfl,hv⟩ := ZFSet.pair_inj.mp he
exact ⟨hy,hv⟩
· rintro ⟨hx,rfl⟩; exact ⟨x,hx,rfl⟩
refine ⟨hd,⟨?_,?_⟩,?_⟩
· intro z hz
obtain ⟨x,hx,rfl⟩ := (hf z).mp hz
exact ⟨x,hx,_,second_mem_doubleUnion hz,rfl⟩
· intro x hx
have hp := (hpair x _).mpr ⟨hx,rfl⟩
exact ⟨_,second_mem_doubleUnion hp,hp,fun v _ hv => (hpair x v).mp hv |>.2⟩
· intro x hx v _ hv
have he := (hpair x v).mp hv |>.2
subst v
have hA : stageImage (seed R) x f = 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⟩
rw [(hpair y w).mp hw |>.2] at hz
exact ⟨y,hy,hz⟩
· rintro ⟨y,hy,hz⟩
exact ⟨y,hy,_,(hpair y _).mpr ⟨hd x hx y hy,rfl⟩,hz⟩
refine ⟨stage R x,hA ▸ stageImage_mem M hM hT hs (hM d hdM x hx) hfM,?_,rfl⟩
refine ⟨seed_subset_stage R x,?_,?_⟩
· intro z hz
rcases (mem_stage R x z).mp hz with hz|⟨y,hy,hz⟩
· exact Or.inl hz
· have hp := (hpair y _).mpr ⟨hd x hx y hy,rfl⟩
exact Or.inr ⟨y,hy,_,second_mem_doubleUnion hp,hp,hz⟩
· intro y hy w _ hw
rw [(hpair y w).mp hw |>.2]
exact definablePower_subset_stage R hy
end TuringRigidity.RelativeConstructible
end OAI
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