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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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