openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /CodeOperationCertificates.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.SumCertificates | |
| namespace OAI | |
| namespace TuringRigidity.OrdinalCoding | |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalArithmetic | |
| universe u | |
| theorem codePowerAt_of_certificates (M : ZFSet.{u}) (hM : Transitive M) | |
| (hω : ZFSet.omega.{u} ∈ M) (y x : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) | |
| (hc : OmegaPowerCertificates M (e x)) (hy : e y = (Ordinal.omega0 ^ (e x).rank).toZFSet) : | |
| (codePowerAt y x).Sat (M : Set ZFSet) e := by | |
| rw [codePowerAt,SentenceForm.sat_rename] | |
| exact omegaPowerSentence_of_certificates M hM hω _ (fun i => he _) hc hy | |
| theorem codeSumAt_of_certificates (M : ZFSet.{u}) (hM : Transitive M) | |
| (hω : ZFSet.omega.{u} ∈ M) (c a b : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) | |
| (α β : Ordinal.{u}) (ha : e a = α.toZFSet) (hb : e b = β.toZFSet) | |
| (hc : SumCertificates M α β) (hy : e c = (α+β).toZFSet) : | |
| (codeSumAt c a b).Sat (M : Set ZFSet) e := by | |
| rw [codeSumAt,SentenceForm.sat_rename] | |
| exact sumSentence_of_certificates M hM hω _ (fun i => he _) α β ha hb hc hy | |
| theorem codeScaleAt_of_certificates (M : ZFSet.{u}) (hM : Transitive M) | |
| (hω : ZFSet.omega.{u} ∈ M) (twice : Bool) (y x : ℕ) | |
| (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) | |
| (a : Ordinal.{u}) (ha : e x = a.toZFSet) | |
| (hc : twice = true → SumCertificates M a a) (hy : e y = (codeScale twice a).toZFSet) : | |
| (codeScaleAt twice y x).Sat (M : Set ZFSet) e := by | |
| cases twice with | |
| | false => exact hy.trans ha.symm | |
| | true => | |
| change (doubleSentence.rename (fun i => if i = 0 then y else x)).Sat (M : Set ZFSet) e | |
| rw [SentenceForm.sat_rename] | |
| rw [doubleSentence,SentenceForm.sat_rename] | |
| exact sumSentence_of_certificates M hM hω | |
| (fun i => e (if (if i = 0 then 0 else 1) = 0 then y else x)) | |
| (fun i => he _) a a ha ha (hc rfl) | |
| (by simpa only [codeScale,↓reduceIte,Ordinal.mul_two] using hy) | |
| end TuringRigidity.OrdinalCoding | |
| end OAI | |