| import OAI.Computability.DegreeRigidity.Representation.GroundDegreePresentation |
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|
| namespace OAI |
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|
| namespace TuringRigidity.RelativeConstructible |
| open TransitiveNameModel BoundedSetTheory |
| universe u |
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|
| def packedReal : {n : ℕ} → (Fin n → Oracle) → Oracle |
| | 0, _ => FixedArithmetic.zero |
| | n+1, v => join (v 0) (packedReal (fun i : Fin n => v i.succ)) |
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| def realPart (P : Oracle) : ℕ → Oracle |
| | 0 => fun k => P (2*k) |
| | i+1 => realPart (fun k => P (2*k+1)) i |
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| theorem realPart_packed {n : ℕ} (v : Fin n → Oracle) (i : Fin n) : |
| realPart (packedReal v) i = v i := by |
| induction n with |
| | zero => exact Fin.elim0 i |
| | succ n ih => |
| refine Fin.cases ?_ (fun j => ?_) i |
| · funext k |
| simp [realPart,packedReal,join] |
| · have ht : (fun k => packedReal v (2*k+1)) = packedReal (fun j : Fin n => v j.succ) := by |
| funext k |
| simp [packedReal,join,Nat.add_div] |
| change realPart (fun k => packedReal v (2*k+1)) j = v j.succ |
| rw [ht] |
| exact ih _ j |
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|
| theorem packedReal_internal (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) |
| {n : ℕ} (v : Fin n → Oracle) (hv : ∀ i, v i ∈ modelReals M) : |
| packedReal v ∈ modelReals M := by |
| induction n with |
| | zero => exact sourceT_zero_real M hM hT |
| | succ n ih => exact sourceT_real_join M hM hT (hv 0) (ih _ (fun i => hv i.succ)) |
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| end TuringRigidity.RelativeConstructible |
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| end OAI |
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