openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GlobalArithmeticGenericProgram.lean
| import OAI.Computability.DegreeRigidity.Representation.PrescribedArithmeticGenericProgram | |
| import OAI.Computability.DegreeRigidity.Representation.PrescribedFirstUncountable | |
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenGroundGeneric | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open TransitiveNameModel BoundedSetTheory CountableForcing ElementaryModel PersistentRestrictions | |
| open GenericIdentity OracleJump CohenColumnRealName | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| theorem global_arithmetic_generic_program (π : Degree ≃o Degree) : | |
| ∃ (R : Oracle) (p : OracleCode), | |
| degree R = π.symm (degree (jump FixedArithmetic.zero)) ∧ | |
| ∀ Y : Oracle, ArithmeticPrefixForcing.Generic (fun _ => iterate R 5) Y → | |
| RepresentsAt π p (iterate R 5) Y := by | |
| classical | |
| by_contra hnone | |
| push Not at hnone | |
| let d := π.symm (degree (jump FixedArithmetic.zero)) | |
| let C := {R : Oracle // degree R = d} × OracleCode | |
| let : Countable {R : Oracle // degree R = d} := (countable_degree_fiber d).to_subtype | |
| obtain ⟨R₀,hR₀⟩ := degree_surjective d | |
| let : Nonempty C := ⟨(⟨R₀,hR₀⟩,OracleCode.zero)⟩ | |
| have hw : ∀ c : C, ∃ Y : Oracle, | |
| ArithmeticPrefixForcing.Generic (fun _ => iterate c.1.val 5) Y ∧ | |
| ¬ RepresentsAt π c.2 (iterate c.1.val 5) Y := by | |
| intro c | |
| exact hnone c.1.val c.2 c.1.property | |
| choose W hWg hWbad using hw | |
| obtain ⟨e,he⟩ := exists_surjective_nat C | |
| obtain ⟨M,hM,hT,hcount,_,⟨K,hK⟩,hWM,himages,ρ,hρ,_,_,_⟩ := | |
| PrescribedFirstUncountable.prescribed_covering_model_with_omega_one π (fun n => W (e n)) | |
| let := hcount | |
| have hA := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded | |
| hK.2.1 (sourceT_omega_mem M hM hT) | |
| obtain ⟨G,_,hG,hME,_,_,_⟩ := CohenGroundGeneric.exists_extension M (ZFSet.prod K ZFSet.omega) | |
| hM hT hA ⊤ | |
| obtain ⟨R,p,_,hR,_,_,hrep,_⟩ := | |
| prescribed_arithmetic_generic_program π M K hM hT hK himages ρ hρ G hG | |
| let c : C := (⟨R,hR⟩,p) | |
| obtain ⟨n,hn⟩ := he c | |
| have hW : W c ∈ modelReals M := by simpa only [hn] using hWM n | |
| exact hWbad c (hrep (W c) (hME hW) (hWg c)) | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |