import OAI.Computability.DegreeRigidity.Computability.ArithmeticPrefixForcing
import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericalCohenSets
import OAI.Computability.DegreeRigidity.CohenForcing.InternalCohenFilters

namespace OAI


namespace TuringRigidity.GroundArithmeticGeneric
open TransitiveNameModel BoundedSetTheory UniformArithmetic ArithmeticPrefixForcing
open InternalCohen CohenBorelForcing CountableForcing
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder

theorem ground_arithmetic (M : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
    (O : Oracles) (hO : ∀ i, realCode (O i) ∈ M) (G : Oracle)
    (hG : AtomicForcing.GroundGeneric M (pushFilter (realFilter G))) :
    Generic O G := by
  intro D hD hd
  let A := arithmeticReal (fun O n => D O (EncodedForcing.word n)) O
  have hA : realCode A ∈ M := sourceT_arithmetic_comprehension M hM hT hD O hO
  have hspec (s : List Bool) : wordCode s ∈ wordImage A ↔ D O s :=
    wordImage_spec (D O) A (arithmeticReal_true _ O) s
  have hden : Dense {q : Condition | wordCode q.word ∈ wordImage A} := by
    intro q
    obtain ⟨s,hqs,hs⟩ := hd.1 q.word
    exact ⟨⟨s⟩,hqs,(hspec s).mpr hs⟩
  obtain ⟨q,hq,hqD⟩ := (groundGeneric_iff M _).mp hG (wordImage A)
    (wordImage_mem M hM hT A hA) hden
  rw [pull_push] at hq
  exact ⟨q.word,(hspec q.word).mp hqD,hq⟩

end TuringRigidity.GroundArithmeticGeneric

end OAI