| 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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|
