| import OAI.Computability.DegreeRigidity.Representation.GlobalArithmeticGenericProgram |
| import OAI.Computability.DegreeRigidity.CohenForcing.InternalCohenRequirements |
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| namespace OAI |
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| namespace TuringRigidity.ManuscriptMain |
| open TransitiveNameModel BoundedSetTheory CountableForcing GenericIdentity OracleJump |
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| theorem generic_program_representation (π : Degree ≃o Degree) : GenericProgramRepresentation π := by |
| obtain ⟨R,p,_,hrep⟩ := RelativeConstructible.global_arithmetic_generic_program π |
| obtain ⟨M,hM,hT,hcount,_,_,hRM,_,_⟩ := |
| PrescribedFirstUncountable.prescribed_covering_model_with_omega_one π (fun _ => R) |
| let := hcount |
| have hP := sourceT_real_iterate M hM hT (hRM 0) 5 |
| obtain ⟨D,hD,hground⟩ := InternalCohen.ground_family M ⟨realCode.{0} R,hRM 0⟩ |
| refine ⟨D,hD,iterate R 5,p,?_⟩ |
| intro A hA |
| have ha := GroundArithmeticGeneric.ground_arithmetic M hM hT (fun _ => iterate R 5) |
| (fun _ => hP) A ((hground A).mp hA) |
| obtain ⟨ht,hπ⟩ := hrep A ha |
| exact ⟨value p (iterate R 5) A,(total_iff p (iterate R 5) A).mp ht,hπ⟩ |
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| theorem borel_representation : BorelRepresentation := |
| fun π => borel_of_generic_program π (generic_program_representation π) |
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| theorem rigidity : MainTheorem := main_of_representation borel_representation |
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| end TuringRigidity.ManuscriptMain |
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| end OAI |
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