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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 | import OAI.Computability.DegreeRigidity.Constructibility.PersistentLocality
import OAI.Computability.DegreeRigidity.Computability.JumpProperties
namespace OAI
namespace TuringRigidity.ExtensionArithmeticRepresentative
open OracleJump IdealInterpretation IdealLocality PersistentRestrictions PersistentLocality
theorem literal_join_program {I : DegreeIdeal} (σ : I ≃o I) (z x : I)
(hz : z.val = degree (jump FixedArithmetic.zero)) (A R : Oracle)
(hA : degree A = x.val) (hR : degree R = (σ.symm z).val) :
∃ X : Oracle, degree X = (σ x).val ∧ Reduces X (iterate (join A R) 5) ∧
∃ e : OracleCode,
OracleCode.eval (oracleFunction (iterate (join A R) 5)) e = oracleFunction X := by
obtain ⟨X,Y,hX,hY,hXY⟩ := image_arithmetic σ z hz x
have hjoin : degree Y = degree (join A R) := by
change degree Y = degree A ⊔ degree R
rw [hA,hR]; exact hY
have hred : Reduces X (iterate (join A R) 5) :=
TuringReducible.trans hXY (iterate_mono ((degree_eq_iff _ _).mp hjoin).1 5)
exact ⟨X,hX,hred,OracleCode.exists_code (RecursiveIn.iff_nat.mp hred)⟩
theorem extension_literal_join_program (I J : CountableIdeal)
(hIJ : I.carrier ⊆ J.carrier) (ρ : I ≃o I) (σ : J ≃o J)
(he : Extends hIJ ρ σ) (hz : degree (jump FixedArithmetic.zero) ∈ I.carrier)
(A R : Oracle) (hA : degree A ∈ J.carrier)
(hR : degree R = (ρ.symm ⟨degree (jump FixedArithmetic.zero),hz⟩).val) :
∃ X : Oracle, degree X = (σ ⟨degree A,hA⟩).val ∧ Reduces X (iterate (join A R) 5) ∧
∃ e : OracleCode,
OracleCode.eval (oracleFunction (iterate (join A R) 5)) e = oracleFunction X := by
let z : I := ⟨degree (jump FixedArithmetic.zero),hz⟩
let zJ : ideal J := ⟨z.val,hIJ z.property⟩
let x : ideal J := ⟨degree A,hA⟩
have hpre : degree R = ((lift σ).symm zJ).val :=
hR.trans (extends_symm he z).symm
exact literal_join_program (lift σ) zJ x rfl A R rfl hpre
end TuringRigidity.ExtensionArithmeticRepresentative
end OAI
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