| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaPowerCertificates |
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| namespace OAI |
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| namespace TuringRigidity.OrdinalArithmetic |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible |
| universe u |
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| theorem omegaGraph_of_exact_pairs (M d f : ZFSet.{u}) (hd : d.IsOrdinal) |
| (hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (omegaNext x)) |
| (hval : ∀ x ∈ d, (Ordinal.omega0 ^ x.rank).toZFSet ∈ M) : |
| OmegaGraph M d (iterUnion 2 f) f := by |
| have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = omegaNext 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 |
| refine ⟨(Ordinal.omega0 ^ x.rank).toZFSet, |
| hval x hx,?_,ZFSet.isOrdinal_toZFSet _,?_⟩ |
| · refine ⟨?_,?_,?_⟩ |
| · intro z hz; exact (omegaPower_members x.rank z).mpr (Or.inl hz) |
| · intro z hz |
| rcases (omegaPower_members x.rank z).mp hz with hz|⟨y,hy,hz⟩ |
| · exact Or.inl hz |
| · rw [(hd.mem hx).toZFSet_rank_eq] at hy |
| have hp := (hpair y _).mpr ⟨hd.subset_of_mem hx hy,rfl⟩ |
| exact Or.inr ⟨y,hy,_,second_mem_doubleUnion hp,hp,hz⟩ |
| · intro y hy w _ hw z hz |
| rw [(hpair y w).mp hw |>.2] at hz |
| exact (omegaPower_members x.rank z).mpr |
| (Or.inr ⟨y,(hd.mem hx).toZFSet_rank_eq.symm ▸ hy,hz⟩) |
| · rw [omegaNext,Ordinal.opow_add_one,Ordinal.rank_toZFSet] |
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| theorem OmegaGraph.union_value_mem {M d r f : ZFSet.{u}} (hg : OmegaGraph M d r f) |
| {x : ZFSet.{u}} (hx : x ∈ d) : (Ordinal.omega0 ^ x.rank).toZFSet ∈ M := by |
| obtain ⟨v,hv,hfv,_⟩ := hg.2.1.2 x hx |
| obtain ⟨A,hA,hs,_,_⟩ := hg.2.2 x hx v hv hfv |
| exact (omegaStageStep_exact hg hx hs) ▸ hA |
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| end TuringRigidity.OrdinalArithmetic |
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| end OAI |
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