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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | 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
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