openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /CollectionHierarchyRecursion.lean
| 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 | |