openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /CohenForcing /CohenGroundProgram.lean
| 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 | |