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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaModelAxiomTransport
import OAI.Computability.DegreeRigidity.SetModels.SetModelArithmeticClosure
namespace OAI
namespace TuringRigidity.OmegaModelCore.OmegaData
open BoundedSetTheory SetModelReals SetModelSyntax
universe u
noncomputable section
variable {α : Type u} (S : OmegaData α)
def RealCode (a : α) (A : Oracle) : Prop :=
S.mem a (S.power S.omega) ∧ ∀ n, S.mem (S.num n) a ↔ A n = true
theorem realCode_collapse {a : α} {A : Oracle} (ha : S.RealCode a A) :
S.collapse (⟨a,1,ha.1⟩ : S.Core) = realSet A := by
obtain ⟨B,hB,hBc⟩ := S.collapse_real a ha.1
have he : B=A := by
funext n
have hn := (hBc n).trans (ha.2 n)
cases hb : B n <;> cases ha : A n <;> simp_all
exact hB.trans (congrArg realSet he)
theorem represented_real_iff (A : Oracle) :
(∃ a : α, S.RealCode a A) ↔ A ∈ reals S.collapsedModel := by
constructor
· rintro ⟨a,ha⟩
exact (S.mem_collapsedModel _).mpr ⟨⟨a,1,ha.1⟩,(S.realCode_collapse ha).symm⟩
· intro hA
change realSet A ∈ S.collapsedModel at hA
obtain ⟨a,ha⟩ := (S.mem_collapsedModel _).mp hA
have hsub : S.collapse a ⊆ S.collapse S.omegaNode := by
rw [S.collapse_omega,←ha]
exact realSet_subset A
refine ⟨a.val,(S.mem_power _ _).mpr ((S.collapse_subset_iff a S.omegaNode).mp hsub),?_⟩
intro n
change S.Rel (S.numNode n) a ↔ A n = true
rw [←S.collapse_mem_iff,S.collapse_num,←ha,nat_mem_realSet]
theorem omega_arithmetic_comprehension (hP : S.PairingAxiom) (hU : S.UnionAxiom)
(hS : S.SeparationAxiom) {P : UniformArithmetic.Predicate} (hp : UniformArithmetic.Arith P)
(O : UniformArithmetic.Oracles) (hO : ∀ i, ∃ a, S.RealCode a (O i))
(A : Oracle) (hA : ∀ n, P O n ↔ A n = true) : ∃ a, S.RealCode a A := by
apply (S.represented_real_iff A).mpr
exact arithmetic_comprehension (S.collapsed_context hP hU hS) hp O
(fun i => (S.represented_real_iff _).mp (hO i)) A hA
theorem omega_lower (hP : S.PairingAxiom) (hU : S.UnionAxiom) (hS : S.SeparationAxiom)
{A B : Oracle} (hB : ∃ b, S.RealCode b B) (hAB : Reduces A B) : ∃ a, S.RealCode a A :=
(S.represented_real_iff _).mpr (lower_mem (S.collapsed_context hP hU hS)
((S.represented_real_iff _).mp hB) hAB)
theorem omega_jump (hP : S.PairingAxiom) (hU : S.UnionAxiom) (hS : S.SeparationAxiom)
{A : Oracle} (hA : ∃ a, S.RealCode a A) : ∃ b, S.RealCode b (OracleJump.jump A) :=
(S.represented_real_iff _).mpr (jump_mem (S.collapsed_context hP hU hS)
((S.represented_real_iff _).mp hA))
end
end TuringRigidity.OmegaModelCore.OmegaData
end OAI
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