openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /InternalRankRecursion.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalHull | |
| import OAI.Computability.DegreeRigidity.OrdinalCodes.RankGraphConstruction | |
| namespace OAI | |
| namespace TuringRigidity.InternalRank | |
| open TransitiveNameModel BoundedSetTheory | |
| universe u | |
| def rankCertificate : SigmaFormula := .existsSet (.existsSet (.bounded | |
| (.conj (.isHull 1 3 4 5) (Code.graph 1 0 2)))) | |
| theorem realize_rankCertificate (M d q : ZFSet.{u}) (hM : Transitive M) | |
| (hS : Separation M) (hdM : d ∈ M) (hqM : q ∈ M) | |
| (hd : Transitive d) (hq : ∀ c, c ∈ q ↔ c ∈ M ∧ c ⊆ d) | |
| {x f : ZFSet.{u}} (hx : x ∈ d) (hf : f ∈ M) : | |
| rankCertificate.Realize M (cons f (cons x (cons d (cons q (fun _ => d))))) ↔ | |
| ∃ r ∈ M, Graph (hull d q x) r f := by | |
| have hxM := hM d hdM x hx | |
| have matrix (c r : ZFSet.{u}) (hc : c ∈ M) (hr : r ∈ M) : | |
| (Formula.conj (.isHull 1 3 4 5) (Code.graph 1 0 2)).Realize M | |
| (cons r (cons c (cons f (cons x (cons d (cons q (fun _ => d))))))) ↔ | |
| c = hull d q x ∧ Graph c r f := by | |
| have he : ∀ i, (cons r (cons c (cons f (cons x | |
| (cons d (cons q (fun _ => d))))))) i ∈ M := by | |
| intro i | |
| rcases i with _|i; exact hr | |
| rcases i with _|i; exact hc | |
| rcases i with _|i; exact hf | |
| rcases i with _|i; exact hxM | |
| rcases i with _|i; exact hdM | |
| rcases i with _|i; exact hqM | |
| exact hdM | |
| rw [Formula.absolute _ M hM _ he] | |
| simp only [Formula.Eval,Formula.eval_isHull,Code.eval_graph,cons_zero,cons_succ] | |
| exact and_congr (isHull_iff M d q hM hS hdM hqM hd hq hx hc) Iff.rfl | |
| change (∃ c ∈ M, ∃ r ∈ M, _) ↔ _ | |
| constructor | |
| · rintro ⟨c,hc,r,hr,h⟩ | |
| obtain ⟨rfl,hg⟩ := (matrix c r hc hr).mp h | |
| exact ⟨r,hr,hg⟩ | |
| · rintro ⟨r,hr,hg⟩ | |
| have hc := hull_mem M d q hM hS hdM hqM hxM | |
| exact ⟨_,hc,r,hr,(matrix _ r hc hr).mpr ⟨rfl,hg⟩⟩ | |
| theorem internal_rankGraph (M d q : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hS : Separation M) (hR : SigmaReplacement 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, Graph (hull d q x) (iterUnion 2 f) f := by | |
| induction x using ZFSet.inductionOn with | |
| | h x ih => | |
| have hxM := hM d hdM x hx | |
| let e := cons d (cons q (fun _ => d)) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i | |
| rcases i with _|i; exact hdM | |
| rcases i with _|i; exact hqM | |
| exact hdM | |
| have cert (y f : ZFSet.{u}) (hy : y ∈ x) (hf : f ∈ M) : | |
| rankCertificate.Realize M (cons f (cons y e)) ↔ | |
| ∃ r ∈ M, Graph (hull d q y) r f := | |
| realize_rankCertificate M d q hM hS hdM hqM hd hq (hd x hx y hy) hf | |
| obtain ⟨b,hb,hbdef⟩ := hR rankCertificate 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 hM 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, Graph (hull d q y) r f := by | |
| constructor | |
| · intro hf | |
| have hfM := hM 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 hM hU hb | |
| have hg (z : ZFSet.{u}) : z ∈ g ↔ | |
| ∃ y ∈ x, ∃ w ∈ hull d q y, z = ZFSet.pair w (rankSet 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 hM hU hf 2,hfg⟩,(hfg.mem_iff z).mpr ⟨w,hw,hz⟩⟩ | |
| have hvalue : rankImage x g = rankSet x := by | |
| apply ZFSet.ext | |
| intro z | |
| rw [mem_rankImage,mem_rankSet] | |
| 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 : rankSet x ∈ M := hvalue ▸ | |
| rankImage_mem M hM hP hU hS hxM hgM | |
| let f := g ∪ ({ZFSet.pair x (rankSet x)} : ZFSet.{u}) | |
| have hfM : f ∈ M := binary_union_mem M hM hP hU hgM | |
| (singleton_mem M hM hP (orderedPair_mem M hM hP hxM hvM)) | |
| refine ⟨f,hfM,graph_of_mem_iff _ f (hull_transitive d q x hd) ?_⟩ | |
| intro z | |
| change z ∈ g ∪ ({ZFSet.pair x (rankSet 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 hM 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 hM 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⟩ | |
| theorem rankSet_mem_of_container (M d : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) | |
| (hS : Separation M) (hR : SigmaReplacement M) | |
| (hdM : d ∈ M) (hd : Transitive d) | |
| {x : ZFSet.{u}} (hx : x ∈ d) : rankSet x ∈ M := by | |
| obtain ⟨q,hqM,hq⟩ := internal_power M hM hPow hdM | |
| obtain ⟨f,hf,hfg⟩ := internal_rankGraph M d q hM hP hU hS hR | |
| hdM hqM hd hq x hx | |
| obtain ⟨v,hv,hfv,_⟩ := hfg.2.2.1 x (self_mem_hull d q hx) | |
| rw [←hfg.correct x (self_mem_hull d q hx) v hv hfv] | |
| exact hM _ (iterUnion_mem M hM hU hf 2) v hv | |
| end TuringRigidity.InternalRank | |
| end OAI | |