| import OAI.Computability.DegreeRigidity.CohenForcing.InternalCohenTailTruth |
| import OAI.Computability.DegreeRigidity.CohenForcing.InternalCohenProjectionSymmetry |
| import OAI.Computability.DegreeRigidity.Constructibility.UniformDefinitionSymmetry |
| import OAI.Computability.DegreeRigidity.Effective.UniformDegreeUniverseName |
| import OAI.Computability.DegreeRigidity.Syntax.BoundedAutomorphismOutput |
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenNiceNameConstruction |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.BoundedForcing |
| open RecursiveNames TransitiveNameModel BoundedSetTheory CountableForcing AtomicForcing |
| open RelativeConstructible CohenGroundPoset InternalCohenRestriction InternalCohenProjectedGeneric |
| open InternalCohenProjectionSymmetry InternalCohenFactor |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder |
| CohenNiceNameConstruction.cohenTop |
|
|
| theorem defined_output_tail_cone (M C B : ZFSet.{0}) |
| (hM : Transitive M) (hT : SourceT M) (hct : (M : Set ZFSet.{0}).Countable) |
| (hC : C ∈ M) (hB : B ∈ M) (hBC : B ⊆ C) |
| (δ : Ordinal.{0}) (Z : ZFSet.{0}) (φ : ElementaryModel.SentenceForm) |
| (f x y : Name (Conditions (conditions C))) |
| (hf : f.encode (label (conditions C)) ∈ M) |
| (hx : x.encode (label (conditions C)) ∈ M) |
| (hy : y.encode (label (conditions C)) ∈ M) |
| (hfv : ∀ G : GenericFilter (Conditions (conditions C)), GroundGeneric M G → |
| f.val G.carrier = definedSubset |
| (level (groundReals (genericExtensionSet M (conditions C) G.carrier)) δ) |
| φ (fun _ : Fin φ.bound => Z)) |
| (hxfix : ∀ a : Conditions (conditions C) ≃o Conditions (conditions C), |
| (∀ s, restrict B (label _ (a s)) = restrict B (label _ s)) → |
| ∀ G : GenericFilter (Conditions (conditions C)), |
| x.val (AutomorphismName.mapFilter a G).carrier = x.val G.carrier) |
| (hyfix : ∀ a : Conditions (conditions C) ≃o Conditions (conditions C), |
| (∀ s, restrict B (label _ (a s)) = restrict B (label _ s)) → |
| ∀ G : GenericFilter (Conditions (conditions C)), |
| y.val (AutomorphismName.mapFilter a G).carrier = y.val G.carrier) |
| (G : GenericFilter (Conditions (conditions C))) (hG : GroundGeneric M G) |
| (houtput : ∃ d ∈ degreeUniverse (groundReals (genericExtensionSet M (conditions C) G.carrier)), |
| ∃ b ∈ degreeUniverse (groundReals (genericExtensionSet M (conditions C) G.carrier)), |
| x.val G.carrier ∈ d ∧ y.val G.carrier ∈ b ∧ ZFSet.pair d b ∈ f.val G.carrier) |
| (p₀ : Conditions (conditions C)) (hp₀ : p₀ ∈ G.carrier) : |
| ∃ p ∈ G.carrier, p ≤ p₀ ∧ |
| ∀ H : GenericFilter (Conditions (conditions C)), GroundGeneric M H → |
| project C B hBC p ∈ (projected C B hBC H).carrier → |
| ∃ d ∈ degreeUniverse (groundReals (genericExtensionSet M (conditions C) H.carrier)), |
| ∃ b ∈ degreeUniverse (groundReals (genericExtensionSet M (conditions C) H.carrier)), |
| x.val H.carrier ∈ d ∧ y.val H.carrier ∈ b ∧ ZFSet.pair d b ∈ f.val H.carrier := by |
| have hc := conditions_mem M C hM hT hC |
| have ho := InternalCollapse.orderSet_mem M hM hT hc |
| have ht (H : GenericFilter (Conditions (conditions C))) : ⊤ ∈ H.carrier := by |
| obtain ⟨q,hq⟩ := H.nonempty; exact H.upper le_top hq |
| obtain ⟨D,hD,hDv⟩ := uniform_degreeUniverse_name M hM hT hc ho (InternalCollapse.orderSet_pair _) |
| let e := push f (push x (push y (fun _ => D))) |
| have he (i : ℕ) : (e i).encode (label (conditions C)) ∈ M := by |
| rcases i with _|_|_|i |
| exact hf; exact hx; exact hy; exact hD |
| have hev (H : GenericFilter (Conditions (conditions C))) : (fun i => (e i).val H.carrier) = |
| cons (f.val H.carrier) (cons (x.val H.carrier) (cons (y.val H.carrier) (fun _ => D.val H.carrier))) := by |
| funext i; rcases i with _|_|_|i <;> rfl |
| have hinv : ∀ w ∈ M, ∀ a : Conditions (conditions C) ≃o Conditions (conditions C), |
| (∀ s t, ZFSet.pair (label _ s) (label _ t) ∈ w ↔ t = a s) → |
| (∀ s, restrict B (label _ (a s)) = restrict B (label _ s)) → |
| ∀ H : GenericFilter (Conditions (conditions C)), GroundGeneric M H → |
| (boundedOutputFormula.Eval (fun i => (e i).val (AutomorphismName.mapFilter a H).carrier) ↔ |
| boundedOutputFormula.Eval (fun i => (e i).val H.carrier)) := by |
| intro w hw a ha hfix H hH |
| have hHa := InternalOrderIsoTransport.map_ground_generic M _ _ w hM hT hc hc hw a ha H hH |
| have hExt := InternalOrderIsoTransport.extension_eq M _ _ w hM hT hc hc hw a ha H hH |
| rw [hev _,hev H,hxfix a hfix H,hyfix a hfix H,hfv _ hHa,hfv H hH, hDv _ hHa (ht _),hDv H hH (ht H),hExt] |
| have hφ : boundedOutputFormula.Eval (fun i => (e i).val G.carrier) := by |
| rw [hev G,boundedOutputFormula_raw,hDv G hG (ht G)] |
| exact houtput |
| obtain ⟨r,hr,hrφ⟩ := (internal_bounded_truth M hM hT.pairing hT.union hT.powerSet |
| hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc ho |
| (InternalCollapse.orderSet_pair _) G hG boundedOutputFormula e he).mp hφ |
| obtain ⟨p,hp,hp₀',hpr⟩ := G.directed hp₀ hr |
| refine ⟨p,hp,hp₀',?_⟩ |
| intro H hH hpH |
| have hv := cone_truth_of_internal_tail_symmetry M C B hM hT hct hC hB hBC |
| boundedOutputFormula e he hinv p (forces_mono _ e hpr hrφ) H hH |
| ((projected_cone_iff C B hBC H p).mp hpH) |
| rw [hev H,boundedOutputFormula_raw,hDv H hH (ht H)] at hv |
| exact hv |
|
|
| end TuringRigidity.BoundedForcing |
|
|
| end OAI |
|
|
