| 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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