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8df23f4 | 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 | import OAI.Computability.DegreeRigidity.CohenForcing.CohenGroundJump
import OAI.Computability.DegreeRigidity.Effective.UniformCohenIteration
import OAI.Computability.DegreeRigidity.Effective.UniformTableSimulation
namespace OAI
namespace TuringRigidity.InternalCohen
open TransitiveNameModel BoundedSetTheory CountableForcing OracleJump
open TableIndices IndexMatrix UniformPrograms
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder
CohenNiceNameConstruction.cohenTop
theorem groundGeneric_iterate_program (M : ZFSet.{0}) (hM : Transitive M)
(hT : SourceT M) (R : Oracle) (hR : realCode R ∈ M) (k : ℕ) :
∃ p : OracleCode, ∀ A : Oracle,
AtomicForcing.GroundGeneric M (pushFilter (CohenBorelForcing.realFilter A)) →
OracleCode.eval (oracleFunction (join A (iterate R k))) p =
oracleFunction (iterate (join A R) k) := by
obtain ⟨p,hp⟩ := UniformCohenJump.iterate_generic_join_program R k
exact ⟨p,fun A hA => hp A (fun j _ => groundGeneric_iterate_oneGeneric M hM hT A R hR hA j)⟩
theorem groundGeneric_table_simulation (M : ZFSet.{0}) (hM : Transitive M)
(hT : SourceT M) (R : Oracle) (hR : realCode R ∈ M) (k d : ℕ) :
∃ e : ℕ, ∀ A : Oracle,
AtomicForcing.GroundGeneric M (pushFilter (CohenBorelForcing.realFilter A)) →
∀ X : Oracle, Represents (iterate (join A R) k) (machine d) X →
Represents (join A (iterate R k)) (machine e) X := by
let S := {A : Oracle //
AtomicForcing.GroundGeneric M (pushFilter (CohenBorelForcing.realFilter A))}
obtain ⟨p,hp⟩ := groundGeneric_iterate_program M hM hT R hR k
have h : Runs (fun A : S => oracleFunction (join A.val (iterate R k)))
(fun A => oracleFunction (iterate (join A.val R) k)) := ⟨p,fun A => hp A.val A.property⟩
obtain ⟨e,he⟩ := h.table_simulation d
exact ⟨e,fun A hA => he ⟨A,hA⟩⟩
theorem selected_column_table_simulation (M K a : ZFSet.{0})
(hM : Transitive M) (hT : SourceT M) (hK : K ∈ M) (ha : a ∈ K)
(R : Oracle) (hR : realCode R ∈ M) (k d : ℕ) :
∃ e : ℕ, ∀ G : GenericFilter (Conditions (CohenColumnRealName.poset K)),
AtomicForcing.GroundGeneric M G → ∀ A : Oracle,
(CohenColumnRealName.realName K a).val G.carrier = realCode A →
∀ X : Oracle, Represents (iterate (join A R) k) (machine d) X →
Represents (join A (iterate R k)) (machine e) X := by
obtain ⟨e,he⟩ := groundGeneric_table_simulation M hM hT R hR k d
exact ⟨e,fun G hG A hv => he A
(CohenColumnRealName.selected_real_prefix_generic M K a hM hT hK ha G hG A hv)⟩
end TuringRigidity.InternalCohen
end OAI
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