openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentLocality.lean
| import OAI.Computability.DegreeRigidity.SetModels.IdealLocality | |
| import OAI.Computability.DegreeRigidity.Constructibility.PersistentRestrictions | |
| import OAI.Computability.DegreeRigidity.SetModels.CountableBound | |
| namespace OAI | |
| namespace TuringRigidity.PersistentLocality | |
| open IdealInterpretation IdealLocality PersistentRestrictions OracleJump | |
| noncomputable section | |
| def ideal (I : CountableIdeal) : DegreeIdeal := | |
| ⟨I.carrier,I.nonempty,fun {_ _} h hmem => I.lower h hmem, | |
| fun {_ _} h₁ h₂ => I.join_mem h₁ h₂⟩ | |
| def bridge (I : CountableIdeal) : I ≃o ideal I where | |
| toFun x := ⟨x.val,x.property⟩ | |
| invFun x := ⟨x.val,x.property⟩ | |
| left_inv _ := rfl | |
| right_inv _ := rfl | |
| map_rel_iff' := Iff.rfl | |
| def lift {I : CountableIdeal} (ρ : I ≃o I) : ideal I ≃o ideal I := | |
| (bridge I).symm.trans (ρ.trans (bridge I)) | |
| def unlift {I : CountableIdeal} (ρ : ideal I ≃o ideal I) : I ≃o I := | |
| (bridge I).trans (ρ.trans (bridge I).symm) | |
| theorem extends_symm {I K : CountableIdeal} {hIK : I.carrier ⊆ K.carrier} | |
| {ρ : I ≃o I} {σ : K ≃o K} (he : Extends hIK ρ σ) : | |
| Extends hIK ρ.symm σ.symm := by | |
| intro x | |
| have h : σ ⟨(ρ.symm x).val,hIK (ρ.symm x).property⟩ = ⟨x.val,hIK x.property⟩ := by | |
| apply Subtype.ext | |
| exact (he (ρ.symm x)).trans (congrArg Subtype.val (ρ.apply_symm_apply x)) | |
| have hh := congrArg (fun y => (σ.symm y).val) h | |
| simpa only [σ.symm_apply_apply] using hh.symm | |
| theorem source_4_1_5 (I J : CountableIdeal) (ρ : I ≃o I) | |
| (hp : Persistent I ρ) (hz : degree (jump FixedArithmetic.zero) ∈ I.carrier) | |
| (hIJ : I.carrier ⊆ J.carrier) (hjump : JumpClosed (ideal J)) : | |
| ∃ σ : J ≃o J, Extends hIJ ρ σ := by | |
| obtain ⟨b,hb⟩ := CountableBound.countable_bounded J.carrier J.countable | |
| obtain ⟨K,hIK,τ,hbK,he⟩ := hp b | |
| have hJK : J.carrier ⊆ K.carrier := fun x hx => K.lower (hb x hx) hbK | |
| let z : I := ⟨degree (jump FixedArithmetic.zero),hz⟩ | |
| let zK : ideal K := ⟨z.val,hIK z.property⟩ | |
| have hforward : ((lift τ) zK).val ∈ J.carrier := by | |
| change (τ ⟨z.val,hIK z.property⟩).val ∈ J.carrier | |
| rw [he z] | |
| exact hIJ (ρ z).property | |
| have hback : ((lift τ).symm zK).val ∈ J.carrier := by | |
| change (τ.symm ⟨z.val,hIK z.property⟩).val ∈ J.carrier | |
| rw [extends_symm he z] | |
| exact hIJ (ρ.symm z).property | |
| obtain ⟨σ,hσ⟩ := source_4_1_4 (lift τ) zK rfl hJK hjump (J.join_mem hforward hback) | |
| refine ⟨unlift σ,fun x => ?_⟩ | |
| exact (hσ ⟨x.val,hIJ x.property⟩).trans (he x) | |
| end | |
| end TuringRigidity.PersistentLocality | |
| end OAI | |