import OAI.Computability.DegreeRigidity.Constructibility.ContextDefinablePower
import OAI.Computability.DegreeRigidity.Constructibility.HierarchyGraph

namespace OAI

namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
theorem definablePower_sigmaDefinable_context (M : ZFSet.{u}) (C : Context M) :
    SigmaDefinable M (fun e => e 0 = definablePower (e 1)) := by
  obtain ⟨Q,hQ,_,hq⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem
  obtain ⟨B,hB,hb⟩ := internal_support_bound C
  have hc := (defSystem_definable C hQ hB 5 3 2 1 0 4).toSigma C.transitive
  have hs := hc.existsSet.existsSet.existsSet.existsSet
  apply hs.congr
  intro e he
  change (∃ G ∈ M, ∃ T ∈ M, ∃ H ∈ M, ∃ Z ∈ M,
    DefSystem Q B (e 1) G T H Z (e 0)) ↔ e 0 = definablePower (e 1)
  constructor
  · rintro ⟨G,_,T,_,H,_,Z,_,h⟩
    exact h.exact hq hb
  · intro h
    rw [h]
    exact internal_defSystem_of_context M C Q B (e 1) (he 1) hq hb

theorem uniform_definable_successor_context (M : ZFSet.{u}) (C : Context M) :
    ∃ p : SigmaFormula, ∃ d : ℕ → ZFSet.{u}, (∀ i, d i ∈ M) ∧
      ∀ A ∈ M, ∀ D ∈ M, p.Realize M (cons D (cons A d)) ↔ D = definablePower A :=
  (definablePower_sigmaDefinable_context M C).binary definablePower

theorem stageImage_mem_context (M : ZFSet.{u}) (C : Context M)
    {s x f : ZFSet.{u}} (hs : s ∈ M) (hx : x ∈ M) (hf : f ∈ M) : stageImage s x f ∈ M := by
  have hr := iterUnion_mem M C.transitive C.union hf 2
  have hd := (((defPairMem C 1 0 4).and (member_definable C 2 0)).existsMem 4).existsMem 1
  let e := cons x (cons f (fun _ => iterUnion 2 f))
  have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hx; exact hf; exact hr
  exact binary_union_mem M C.transitive C.pairing C.union hs
    (hd.sep_mem C e he (union_mem M C.transitive C.union hr))

theorem hierarchyGraph_of_mem_iff_context (M R d f : ZFSet.{u}) (C : Context M)
    (hs : seed R ∈ M) (hdM : d ∈ M) (hfM : f ∈ M) (hd : Transitive d)
    (hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x))) :
    HierarchyGraph M (seed R) d (iterUnion 2 f) f := by
  have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = definablePower (stage R 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
    have hA : stageImage (seed R) x f = stage R x := by
      apply ZFSet.ext; intro z
      rw [mem_stageImage,mem_stage]
      apply or_congr_right
      constructor
      · rintro ⟨y,hy,w,hw,hz⟩
        rw [(hpair y w).mp hw |>.2] at hz
        exact ⟨y,hy,hz⟩
      · rintro ⟨y,hy,hz⟩
        exact ⟨y,hy,_,(hpair y _).mpr ⟨hd x hx y hy,rfl⟩,hz⟩
    refine ⟨stage R x,hA ▸ stageImage_mem_context M C hs (C.transitive d hdM x hx) hfM,?_,rfl⟩
    refine ⟨seed_subset_stage R x,?_,?_⟩
    · intro z hz
      rcases (mem_stage R x z).mp hz with hz|⟨y,hy,hz⟩
      · exact Or.inl hz
      · have hp := (hpair y _).mpr ⟨hd x hx y hy,rfl⟩
        exact Or.inr ⟨y,hy,_,second_mem_doubleUnion hp,hp,hz⟩
    · intro y hy w _ hw
      rw [(hpair y w).mp hw |>.2]
      exact definablePower_subset_stage R hy

end TuringRigidity.RelativeConstructible

end OAI