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