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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 | import OAI.Computability.DegreeRigidity.SetModels.RelationPresentation
import OAI.Computability.DegreeRigidity.Computability.OracleJump
namespace OAI
namespace TuringRigidity.DecodingTheorem
open IndexPresentation IndexTuples ArithmeticHierarchy RelationPresentation
open SetCoding RelationCoding BoundedDecoding OracleJump
def Presented {n} (Y : Oracle) (p : RelationCode n) (t : ℕ) : Prop :=
(∀ i : Fin n, Dom Y (entry i.val t)) ∧ p.Holds (fun i => value Y (entry i.val t))
theorem presentation_exact {n} (Y : Oracle) (p : RelationCode n)
(hp : RelationBelow p (degree Y)) (v : Fin n → Degree) :
p.Holds v ↔ ∃ t, Presented Y p t ∧ ∀ i, value Y (entry i.val t) = v i := by
constructor
· intro hv
have hb : ∀ i, v i ≤ degree Y := by
obtain ⟨c,hc,_⟩ := hv
exact fun i => (graph_bounded (hp.1 i) (hc i)).1
choose e he hev using (fun i => value_surjective Y (v i) (hb i))
obtain ⟨t,ht⟩ := tuple_surjective e
have hv' : (fun i : Fin n => value Y (entry i.val t)) = v := by
funext i
rw [ht i, hev i]
exact ⟨t, ⟨fun i => (ht i) ▸ he i, hv' ▸ hv⟩, fun i => congrFun hv' i⟩
· rintro ⟨t,⟨_,ht⟩,hv⟩
exact (funext hv) ▸ ht
theorem sigma5_normal_form {Y P} (h : Sigma Y 5 P) :
∃ M : ℕ → Prop, RecursivePred Y M ∧ ∀ x,
P x ↔ ∃ a, ∀ b, ∃ c, ∀ d, ∃ e,
M (Nat.pair (Nat.pair (Nat.pair (Nat.pair (Nat.pair x a) b) c) d) e) := by
obtain ⟨A,hA,eA⟩ := h
obtain ⟨B,hB,eB⟩ := hA
obtain ⟨C,hC,eC⟩ := hB
obtain ⟨D,hD,eD⟩ := hC
obtain ⟨M,hM,eM⟩ := hD
refine ⟨M,hM,fun x => ?_⟩
simp only [eA, eB, eC, eD, eM, Quant, Bool.not_true, Bool.not_false, Bool.false_eq_true, ↓reduceIte]
theorem decoding_theorem {n} (Y : Oracle) (R : Set (Fin n → Degree)) (p : RelationCode n)
(hp : RelationBelow p (degree Y)) (hcode : ∀ v, p.Holds v ↔ v ∈ R) :
∃ P : ℕ → Prop, Sigma Y 5 P ∧ RecursivePred (iterate Y 5) P ∧
∀ v, v ∈ R ↔ ∃ t, P t ∧ ∀ i, value Y (entry i.val t) = v i := by
have h := sigma5_decoding Y p hp
refine ⟨Presented Y p, h, form_recursive h, fun v => ?_⟩
exact (hcode v).symm.trans (presentation_exact Y p hp v)
end TuringRigidity.DecodingTheorem
end OAI
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