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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 50 51 52 53 54 55 56 57 58 59 60 | import OAI.Computability.DegreeRigidity.SetModels.ClosedPaddingCertificates
namespace OAI
namespace TuringRigidity.OrdinalCoding
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalArithmetic
universe u
theorem cofinal_padded_code_certificates (M : ZFSet.{u}) (hM : Transitive M)
(hT : SourceT M) {n : ℕ} (v : Fin n → Ordinal.{u})
(hv : ∀ i, (v i).toZFSet ∈ M) (θ : Ordinal.{u}) (hθ : θ.toZFSet ∈ M) :
∃ β : Ordinal.{u}, β.toZFSet ∈ M ∧ θ < β ∧ realSeedOffset < β ∧
(heightDomainIndex (paddedCode v β)).toZFSet ∈ M ∧
PaddedCodeCertificates (level (groundReals M) (heightDomainIndex (paddedCode v β))) v β := by
obtain ⟨τ,hτ,homega⟩ := internal_explicit_omega_threshold M hM hT
obtain ⟨η,hη,hvec⟩ := vectorCode_certificates_at_levels M hM hT v hv
have hk := vectorCode_internal M hM hT v hv
have hoffM : realSeedOffset.toZFSet ∈ M := by
rw [← ground_seed_offset M hM hT]
exact ordinalHeight_mem_model M _ hM hT.separation.finitePrefix.bounded
(seed_mem M _ hM hT (groundReals_mem M hM hT)) (seed_transitive _)
let γ := max θ (max τ (max η (max realSeedOffset (vectorCode v))))
have hγ : γ.toZFSet ∈ M := internal_ordinal_max M hθ
(internal_ordinal_max M hτ (internal_ordinal_max M hη (internal_ordinal_max M hoffM hk)))
obtain ⟨β,hβM,hγβ,hβ,hfix⟩ := cofinal_internal_omega_fixed_points M hM hT γ hγ
have hθβ : θ < β := (le_max_left _ _).trans_lt hγβ
have htail : max τ (max η (max realSeedOffset (vectorCode v))) < β :=
(le_max_right _ _).trans_lt hγβ
have hτβ : τ < β := (le_max_left _ _).trans_lt htail
have htail' : max η (max realSeedOffset (vectorCode v)) < β := (le_max_right _ _).trans_lt htail
have hηβ : η < β := (le_max_left _ _).trans_lt htail'
have hlast : max realSeedOffset (vectorCode v) < β := (le_max_right _ _).trans_lt htail'
have hoffβ : realSeedOffset < β := (le_max_left _ _).trans_lt hlast
have hkβ : vectorCode v < β := (le_max_right _ _).trans_lt hlast
have hoff : realSeedOffset+β=β := by
have h := Ordinal.add_omega0_opow (a := realSeedOffset) (b := β) (by rw [hfix]; exact hoffβ)
simpa only [hfix] using h
have hpow := lower_omega_closure_of_fixed_point M τ β homega hβ hτβ hfix
have hc := padded_certificates_of_lower_closure M hM hT v β hβ hoff hkβ
(hvec β hηβ.le).2 hpow
exact ⟨β,hβM,hθβ,hoffβ,padded_height_index_from_inputs M hM hT v β hv hβM,hc⟩
theorem cofinal_largest_entry_decoder (M : ZFSet.{u}) (hM : Transitive M)
(hT : SourceT M) {n : ℕ} (v : Fin n → Ordinal.{u})
(hv : ∀ i, (v i).toZFSet ∈ M) (θ : Ordinal.{u}) (hθ : θ.toZFSet ∈ M) :
∃ β : Ordinal.{u}, β.toZFSet ∈ M ∧ θ < β ∧ realSeedOffset < β ∧
(heightDomainIndex (paddedCode v β)).toZFSet ∈ M ∧
∀ (k : Fin n) (e : ℕ → ZFSet.{u}),
e 0 ∈ level (groundReals M) (heightDomainIndex (paddedCode v β)) →
((largestPaddedEntrySentence n k).Sat
(level (groundReals M) (heightDomainIndex (paddedCode v β)) : Set ZFSet) e ↔
e 0 = (v k).toZFSet) := by
obtain ⟨β,hβ,hθβ,hoff,hidx,hcert⟩ := cofinal_padded_code_certificates M hM hT v hv θ hθ
exact ⟨β,hβ,hθβ,hoff,hidx,fun k e he =>
largestPaddedEntrySentence_of_certificates M hM hT n k v β hoff.le e he hcert⟩
end TuringRigidity.OrdinalCoding
end OAI
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