File size: 2,614 Bytes
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 63 64 65 | import OAI.Computability.DegreeRigidity.Constructibility.PersistentPresentation
namespace OAI
namespace TuringRigidity.PersistentCountability
open EncodedForcing OracleJump ArithmeticHierarchy PersistentRestrictions PersistentPresentation
noncomputable section
attribute [local instance] Classical.propDecidable
theorem presentation_exists (I : CountableIdeal) : ∃ H, Presented I H := by
obtain ⟨f,hf⟩ := I.countable.exists_eq_range I.nonempty
choose A hA using (fun n => degree_surjective (f n))
refine ⟨CountableBound.sum A,fun x => ?_⟩
have hc (k : ℕ) : columns (CountableBound.sum A) k = A k := by
funext n
simp only [columns,CountableBound.sum,Nat.unpair_pair]
simp only [hc,hA,hf,Set.mem_range]
def graphOracle {I : CountableIdeal} {H : Oracle} (hpres : Presented I H)
(ρ : I ≃o I) : Oracle := fun n => decide (Graph ρ hpres n)
theorem graphOracle_injective {I : CountableIdeal} {H : Oracle}
(hpres : Presented I H) : Function.Injective (graphOracle hpres) := by
intro ρ σ he
apply OrderIso.ext
funext x
obtain ⟨n,hn⟩ := (hpres x.val).mp x.property
obtain ⟨j,hj⟩ := (hpres (ρ x).val).mp (ρ x).property
have hx : entry hpres n = x := Subtype.ext hn
have hp : Graph ρ hpres (Nat.pair n j) := by
simp only [PersistentPresentation.Graph,Nat.unpair_pair,hx,hj]
have hq : Graph σ hpres (Nat.pair n j) := by
by_contra hn
have hh := congrFun he (Nat.pair n j)
simp only [graphOracle,decide_eq_true hp,decide_eq_false hn] at hh
cases hh
apply Subtype.ext
simpa only [PersistentPresentation.Graph,Nat.unpair_pair,hx,hj] using hq.symm
theorem graphOracle_reduces {I : CountableIdeal} {H Y : Oracle}
(hpres : Presented I H) (ρ : I ≃o I) (h : RecursivePred Y (Graph ρ hpres)) :
Reduces (graphOracle hpres ρ) Y := by
apply RecursiveIn.iff_nat.mpr
exact h.of_eq (fun n => by simp [oracleFunction,graphOracle])
theorem arithmetic_maps_countable {I : CountableIdeal} {H : Oracle}
(hpres : Presented I H) (Y : Oracle) :
{ρ : I ≃o I | RecursivePred Y (Graph ρ hpres)}.Countable := by
apply ((countable_reduces Y).preimage (graphOracle_injective hpres)).mono
intro ρ hρ
exact graphOracle_reduces hpres ρ hρ
theorem source_4_1_9 (I : CountableIdeal)
(hz : degree (jump FixedArithmetic.zero) ∈ I.carrier) :
{ρ : I ≃o I | Persistent I ρ}.Countable := by
obtain ⟨H,hpres⟩ := presentation_exists I
apply (arithmetic_maps_countable hpres (iterate H 11)).mono
intro ρ hp
exact (source_4_1_8 ρ hp hz hpres).2
end
end TuringRigidity.PersistentCountability
end OAI
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