openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /InternalOmegaPower.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaPowerGraph | |
| namespace OAI | |
| namespace TuringRigidity.OrdinalArithmetic | |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| theorem internal_omegaGraph (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| (x : ZFSet.{u}) (hxM : x ∈ M) (hxo : x.IsOrdinal) : | |
| ∃ f ∈ M, OmegaGraph M (insert x x) (iterUnion 2 f) f := by | |
| have h1 : (1 : Ordinal.{u}).toZFSet ∈ M := by | |
| rw [← Nat.cast_one,toZFSet_nat] | |
| exact hM _ (omega_mem M hM hT.separation.finitePrefix.bounded hT.infinity) _ ((mem_omega _).mpr ⟨1,rfl⟩) | |
| obtain ⟨p,e,he,hp⟩ := uniform_omega_certificate M hM hT | |
| induction x using ZFSet.inductionOn with | |
| | h x ih => | |
| have cert (y f : ZFSet.{u}) (hy : y ∈ x) (hf : f ∈ M) := hp y (hM x hxM y hy) f hf | |
| obtain ⟨b,hb,hbdef⟩ := hT.replacement.finitePrefix p e he x hxM (by | |
| intro y hy | |
| obtain ⟨f,hf,hfg⟩ := ih y hy (hM x hxM y hy) (hxo.mem hy) | |
| refine ⟨f,hf,(cert y f hy hf).mpr | |
| ⟨_,iterUnion_mem M hM hT.union 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, OmegaGraph M (insert y 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 (hM x hxM y hy) (hxo.mem 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 hT.union hb | |
| have hg (z : ZFSet.{u}) : z ∈ g ↔ ∃ y ∈ x, z = ZFSet.pair y (omegaNext y) := by | |
| constructor | |
| · intro hz | |
| obtain ⟨f,hf,hzf⟩ := ZFSet.mem_sUnion.mp hz | |
| obtain ⟨y,hy,r,_,hfg⟩ := (hbg f).mp hf | |
| obtain ⟨w,hw,hz⟩ := (hfg.mem_iff z).mp hzf | |
| rcases ZFSet.mem_insert_iff.mp hw with rfl|hw | |
| · exact ⟨w,hy,hz⟩ | |
| · exact ⟨w,hxo.subset_of_mem hy hw,hz⟩ | |
| · rintro ⟨y,hy,hz⟩ | |
| obtain ⟨f,hf,hfg⟩ := ih y hy (hM x hxM y hy) (hxo.mem hy) | |
| exact ZFSet.mem_sUnion.mpr ⟨f,(hbg f).mpr | |
| ⟨y,hy,_,iterUnion_mem M hM hT.union hf 2,hfg⟩, | |
| (hfg.mem_iff z).mpr ⟨y,ZFSet.mem_insert_iff.mpr (Or.inl rfl),hz⟩⟩ | |
| have hvalue : stageImage (1 : Ordinal.{u}).toZFSet x g = | |
| (Ordinal.omega0 ^ x.rank).toZFSet := by | |
| apply ZFSet.ext; intro z | |
| rw [mem_stageImage,omegaPower_members,hxo.toZFSet_rank_eq] | |
| apply or_congr_right | |
| constructor | |
| · rintro ⟨y,hy,w,hfw,hz⟩ | |
| obtain ⟨v,_,hp⟩ := (hg _).mp hfw | |
| obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp hp | |
| exact ⟨y,hy,hz⟩ | |
| · rintro ⟨y,hy,hz⟩ | |
| exact ⟨y,hy,_,(hg _).mpr ⟨y,hy,rfl⟩,hz⟩ | |
| have hpow : (Ordinal.omega0 ^ x.rank).toZFSet ∈ M := | |
| hvalue ▸ stageImage_mem M hM hT h1 hxM hgM | |
| have hvM : omegaNext x ∈ M := by | |
| rw [omegaNext,Ordinal.opow_add_one] | |
| exact ordinal_mul_omega_internal M hM hT _ hpow | |
| let f := g ∪ ({ZFSet.pair x (omegaNext x)} : ZFSet.{u}) | |
| have hfM : f ∈ M := binary_union_mem M hM hT.pairing hT.union hgM | |
| (singleton_mem M hM hT.pairing (orderedPair_mem M hM hT.pairing hxM hvM)) | |
| have hsucc : (insert x x).IsOrdinal := by | |
| rw [← hxo.toZFSet_rank_eq,← Ordinal.toZFSet_add_one] | |
| exact ZFSet.isOrdinal_toZFSet _ | |
| refine ⟨f,hfM,omegaGraph_of_mem_iff M _ f hM hT | |
| (insert_self_mem M hM hT.pairing hT.union hxM) hfM hsucc ?_⟩ | |
| intro z | |
| change z ∈ g ∪ ({ZFSet.pair x (omegaNext x)} : ZFSet.{u}) ↔ _ | |
| rw [ZFSet.mem_union,ZFSet.mem_singleton,hg] | |
| constructor | |
| · rintro (⟨y,hy,hz⟩|hz) | |
| · exact ⟨y,ZFSet.mem_insert_of_mem _ hy,hz⟩ | |
| · exact ⟨x,ZFSet.mem_insert_iff.mpr (Or.inl rfl),hz⟩ | |
| · rintro ⟨y,hy,hz⟩ | |
| rcases ZFSet.mem_insert_iff.mp hy with rfl|hy | |
| · exact Or.inr hz | |
| · exact Or.inl ⟨y,hy,hz⟩ | |
| theorem ordinal_omega_opow_internal (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| (a : Ordinal.{u}) (ha : a.toZFSet ∈ M) : (Ordinal.omega0 ^ a).toZFSet ∈ M := by | |
| obtain ⟨f,hf,hg⟩ := internal_omegaGraph M hM hT a.toZFSet ha (ZFSet.isOrdinal_toZFSet a) | |
| have h1 : (1 : Ordinal.{u}).toZFSet ∈ M := by | |
| rw [← Nat.cast_one,toZFSet_nat] | |
| exact hM _ (omega_mem M hM hT.separation.finitePrefix.bounded hT.infinity) _ ((mem_omega _).mpr ⟨1,rfl⟩) | |
| have hv := omegaStageImage _ f a.toZFSet hg.1 (ZFSet.mem_insert_iff.mpr (Or.inl rfl)) hg.mem_iff | |
| rw [Ordinal.rank_toZFSet] at hv | |
| exact hv ▸ stageImage_mem M hM hT h1 ha hf | |
| end TuringRigidity.OrdinalArithmetic | |
| end OAI | |