openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /ClosedPaddingCertificates.lean
| import OAI.Computability.DegreeRigidity.SetModels.RecursiveOmegaBound | |
| import OAI.Computability.DegreeRigidity.Constructibility.InternalOmegaFixedPoint | |
| import OAI.Computability.DegreeRigidity.OrdinalCodes.VectorCodeLevels | |
| namespace OAI | |
| namespace TuringRigidity.OrdinalCoding | |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalArithmetic | |
| universe u | |
| theorem padded_certificates_of_lower_closure (M : ZFSet.{u}) (hM : Transitive M) | |
| (hT : SourceT M) {n : ℕ} (v : Fin n → Ordinal.{u}) (β : Ordinal.{u}) | |
| (hβ : Order.IsSuccLimit β) (hoff : realSeedOffset+β=β) (hv : vectorCode v < β) | |
| (hvec : VectorCodeCertificates (level (groundReals M) β) v) | |
| (hpow : ∀ c < β, (Ordinal.omega0 ^ c).toZFSet ∈ level (groundReals M) β ∧ | |
| OmegaPowerCertificates (level (groundReals M) β) c.toZFSet) : | |
| PaddedCodeCertificates (level (groundReals M) (heightDomainIndex (paddedCode v β))) v β := by | |
| let R := groundReals M | |
| let c := paddedCode v β | |
| let δ := heightDomainIndex c | |
| have hβpos : 0 < β := hβ.bot_lt | |
| have hβc : β < c := padding_lt_paddedCode v β | |
| have hcodeabs := offset_absorbed_above realSeedOffset β c hoff hβc.le | |
| have hδ : δ=c+1 := heightDomainIndex_of_absorbed c hcodeabs | |
| have hβδ : β ≤ δ := hβc.le.trans (hδ ▸ le_self_add) | |
| have hcut : ordinalHeight (seed R)+c=c := by rw [ground_seed_offset M hM hT]; exact hcodeabs | |
| have hmem (a : Ordinal.{u}) (ha : a ≤ c) : a.toZFSet ∈ level R δ := by | |
| rw [hδ,ordinal_mem_level_iff,← add_assoc,hcut] | |
| exact Order.lt_add_one_iff.mpr ha | |
| have hb1 : β+1 ≤ c := Order.add_one_le_iff.mpr hβc | |
| have hu : Ordinal.omega0 ^ (β+1) < c := by | |
| change Ordinal.omega0 ^ (β+1) < pairCode β (vectorCode v) | |
| rw [pairCode,ite_eq_left hv.le] | |
| simpa only [add_zero] using (add_lt_add_iff_left (Ordinal.omega0 ^ (β+1))).mpr | |
| (Ordinal.opow_pos (vectorCode v) Ordinal.omega0_pos) | |
| have hvpow := level_mono R hβδ (hpow (vectorCode v) hv).1 | |
| have hcβ := padded_omega_boundary_certificates M hM hT v β hβ hoff hv.le hpow | |
| (omega_boundary_product_without_offset M hM hT β hβpos) | |
| have hck := (hpow (vectorCode v) hv).2.mono (level_mono R hβδ) | |
| have hsum := padded_final_sum_certificates M hM hT v β hβpos hoff hv.le | |
| refine ⟨hmem β hβc.le,hmem (vectorCode v) (hv.trans hβc).le,?_,hvec.mono (level_mono R hβδ)⟩ | |
| rw [PairCodeCertificates,ite_eq_left hv.le] | |
| exact ⟨hmem (β+1) hb1,hmem _ hu.le,hvpow,hmem _ hu.le,hcβ,hck, | |
| fun h => Bool.noConfusion h,hsum⟩ | |
| end TuringRigidity.OrdinalCoding | |
| end OAI | |