openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Representation /GroundArithmeticGeneric.lean
| 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 | |