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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 54 55 56 57 58 59 60 61 62 63 | 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
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