File size: 3,106 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 | import OAI.Computability.DegreeRigidity.Effective.GuardedExtraction
import OAI.Computability.DegreeRigidity.Computability.NormalizedRecovery
import OAI.Computability.DegreeRigidity.Effective.AntichainParameters
namespace OAI
namespace TuringRigidity.GuardedCoding
open EncodedForcing StageNormalization AntichainParameters
theorem effective_coding (A : Oracle) : ∃ G₀ G₁ : Oracle,
Reduces (join G₀ G₁) (OracleJump.jump A) ∧
(∀ k, ∃ C : Oracle, Reduces C (join G₀ (columns A k)) ∧
Reduces C (join G₁ (columns A k)) ∧ ¬ Reduces C (columns A k)) ∧
∀ Y Z : Oracle, Reduces Y A → Reduces Z (join G₀ Y) →
Reduces Z (join G₁ Y) → ¬ Reduces Z Y → ∃ k, Reduces (columns A k) Y := by
let B := normalized A
obtain ⟨hbound,hfirst⟩ := GuardedLimits.effective_first_clause B
(fun k => (normalized_spec A k).2.1)
refine ⟨GuardedLimits.G₀ B,GuardedLimits.G₁ B,
reduces_trans hbound (OracleJump.jump_mono (normalized_reduces A)),?_,?_⟩
· intro k
obtain ⟨C,h₀,h₁,hn⟩ := hfirst k
obtain ⟨hBA,hAB⟩ := (degree_eq_iff _ _).mp (normalized_spec A k).1
exact ⟨C,reduces_trans h₀ (join_mono (reduces_refl _) hBA),
reduces_trans h₁ (join_mono (reduces_refl _) hBA),fun h => hn (reduces_trans h hAB)⟩
· intro Y Z hY h₀ h₁ hn
obtain ⟨k,hk⟩ := GuardedExtraction.common_lower_computes_column
(fun k => (normalized_spec A k).2.2)
(reduces_trans hY (NormalizedRecovery.normalized_inverse A))
(reduces_trans h₀ (AntichainCoding.swap_join_reduces _ _))
(reduces_trans h₁ (AntichainCoding.swap_join_reduces _ _)) hn
exact ⟨k,reduces_trans (((degree_eq_iff _ _).mp (normalized_spec A k).1).2) hk⟩
theorem effective_antichain_parameters (A : Oracle)
(hAnti : ∀ i j, degree (columns A i) ≤ degree (columns A j) →
degree (columns A i) = degree (columns A j)) :
∃ g₀ g₁ : Degree, g₀ ⊔ g₁ ≤ OracleJump.degreeJump (degree A) ∧
∀ x : Degree, CodedAntichain (degree A) g₀ g₁ x ↔ ∃ k, degree (columns A k) = x := by
obtain ⟨G₀,G₁,hbound,hfirst,hsecond⟩ := effective_coding A
have hm : ∀ k, NontrivialCommonLower (degree A) (degree G₀) (degree G₁) (degree (columns A k)) := by
intro k
obtain ⟨C,h₀,h₁,hn⟩ := hfirst k
exact ⟨CodingExtraction.column_projection_reduces A k,degree C,h₀,h₁,hn⟩
have hc : ∀ x, NontrivialCommonLower (degree A) (degree G₀) (degree G₁) x →
∃ k, degree (columns A k) ≤ x := by
intro x hx
obtain ⟨Y,rfl⟩ := degree_surjective x
obtain ⟨z,h₀,h₁,hn⟩ := hx.2
obtain ⟨Z,rfl⟩ := degree_surjective z
exact hsecond Y Z hx.1 h₀ h₁ hn
refine ⟨degree G₀,degree G₁,hbound,fun x => ?_⟩
constructor
· intro hx
obtain ⟨k,hk⟩ := hc x hx.1
exact ⟨k,le_antisymm hk (hx.2 _ (hm k) hk)⟩
· rintro ⟨k,rfl⟩
refine ⟨hm k,fun y hy hyk => ?_⟩
obtain ⟨j,hj⟩ := hc y hy
exact hAnti j k (hj.trans hyk) ▸ hj
end TuringRigidity.GuardedCoding
end OAI
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