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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaPowerCertificates
namespace OAI
namespace TuringRigidity.OrdinalArithmetic
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible
universe u
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]
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
end TuringRigidity.OrdinalArithmetic
end OAI
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