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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 61 62 | import OAI.Computability.DegreeRigidity.OrdinalCodes.CodeBranchCertificates
import OAI.Computability.DegreeRigidity.OrdinalCodes.SumCertificateLevels
import OAI.Computability.DegreeRigidity.Constructibility.UniformOmegaLevels
namespace OAI
namespace TuringRigidity.OrdinalCoding
open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalArithmetic
universe u
theorem codeBranch_certificates_at_levels (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(twice : Bool) (a b : Ordinal.{u}) (ha : a.toZFSet ∈ M) (hb : b.toZFSet ∈ M) :
∃ γ : Ordinal.{u}, γ.toZFSet ∈ M ∧ ∀ δ : Ordinal.{u}, γ ≤ δ →
a.toZFSet ∈ level (groundReals M) δ ∧ b.toZFSet ∈ level (groundReals M) δ ∧
(codeBranchValue twice a b).toZFSet ∈ level (groundReals M) δ ∧
CodeBranchCertificates (level (groundReals M) δ) twice a b := by
have hk := internal_ordinal_succ M hM hT a ha
have hu := ordinal_omega_opow_internal M hM hT (a+1) hk
have hv := ordinal_omega_opow_internal M hM hT b hb
have ht := codeScale_internal M hM hT twice _ hu
obtain ⟨κ,hκ,hκL⟩ := omega_certificates_at_levels M hM hT (a+1) hk
obtain ⟨ℓ,hℓ,hℓL⟩ := omega_certificates_at_levels M hM hT b hb
obtain ⟨μ,hμ,hμL⟩ := sum_certificates_at_levels M hM hT _ _ hu hu
obtain ⟨ν,hν,hνL⟩ := sum_certificates_at_levels M hM hT _ _ ht hv
refine ⟨max (max κ ℓ) (max μ ν),
internal_ordinal_max M (internal_ordinal_max M hκ hℓ) (internal_ordinal_max M hμ hν),?_⟩
intro δ hδ
have hkδ := (le_max_left κ ℓ).trans ((le_max_left (max κ ℓ) (max μ ν)).trans hδ)
have hlδ := (le_max_right κ ℓ).trans ((le_max_left (max κ ℓ) (max μ ν)).trans hδ)
have hmδ := (le_max_left μ ν).trans ((le_max_right (max κ ℓ) (max μ ν)).trans hδ)
have hnδ := (le_max_right μ ν).trans ((le_max_right (max κ ℓ) (max μ ν)).trans hδ)
obtain ⟨hkL,huL,hpc,_⟩ := hκL δ hkδ
obtain ⟨hbL,hvL,hqc,_⟩ := hℓL δ hlδ
obtain ⟨_,_,_,hdc⟩ := hμL δ hmδ
obtain ⟨htL,_,hcL,hsc⟩ := hνL δ hnδ
have haL := ordinal_le_internal (level_transitive _ _) hkL (lt_add_one a).le
exact ⟨haL,hbL,hcL,hkL,huL,hvL,htL,hpc,hqc,fun _ => hdc,hsc⟩
theorem pairCode_certificates_at_levels (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(a b : Ordinal.{u}) (ha : a.toZFSet ∈ M) (hb : b.toZFSet ∈ M) :
∃ γ : Ordinal.{u}, γ.toZFSet ∈ M ∧ ∀ δ : Ordinal.{u}, γ ≤ δ →
a.toZFSet ∈ level (groundReals M) δ ∧ b.toZFSet ∈ level (groundReals M) δ ∧
(pairCode a b).toZFSet ∈ level (groundReals M) δ ∧
PairCodeCertificates (level (groundReals M) δ) a b := by
by_cases hle : b ≤ a
· obtain ⟨γ,hγ,hL⟩ := codeBranch_certificates_at_levels M hM hT false a b ha hb
refine ⟨γ,hγ,?_⟩
intro δ hδ
simpa only [pairCode,ite_eq_left hle,PairCodeCertificates,codeBranchValue,codeScale,
Bool.false_eq_true,↓reduceIte] using hL δ hδ
· obtain ⟨γ,hγ,hL⟩ := codeBranch_certificates_at_levels M hM hT true b a hb ha
refine ⟨γ,hγ,?_⟩
intro δ hδ
obtain ⟨hbL,haL,hc,hcert⟩ := hL δ hδ
refine ⟨haL,hbL,?_,?_⟩
· simpa only [pairCode,ite_eq_right hle,codeBranchValue,codeScale,↓reduceIte] using hc
· simpa only [PairCodeCertificates,ite_eq_right hle] using hcert
end TuringRigidity.OrdinalCoding
end OAI
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