| import OAI.Computability.DegreeRigidity.Representation.PairGenericSelection |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.GenericFactor |
| open FiniteShuffle ShuffleRequirements PairGenericSelection GenericTopology Set |
|
|
| def Rectangle (D : List Bool → Prop) (s t : List Bool) : Prop := |
| ∀ Y Z, Realizes s Y → Realizes t Z → join Y Z ∈ OpenSet D |
|
|
| theorem rectangle_dense (D : List Bool → Prop) (hD : DenseOpen D) (s t : List Bool) : |
| ∃ u v, s <+: u ∧ t <+: v ∧ Rectangle D u v := by |
| let Y : Oracle := fun i => s.getD i false |
| let Z : Oracle := fun i => t.getD i false |
| let n := max s.length t.length |
| obtain ⟨w,hw,hwD⟩ := hD.1 (initial (join Y Z) (2*n)) |
| let H : Oracle := fun i => w.getD i false |
| have hH : Agree (2*n) (join Y Z) H := agree_of_prefix hw |
| have hlen : 2*n ≤ w.length := by simpa using hw.length_le |
| refine ⟨initial (column false H) w.length,initial (column true H) w.length,?_,?_,?_⟩ |
| · rw [←prefix_default s] |
| apply initial_prefix (n := s.length) (m := w.length) (by dsimp [n] at hlen; omega) |
| intro i hi |
| exact (show Y i = H (2*i) from by simpa using hH (2*i) (by dsimp [n]; omega)) |
| · rw [←prefix_default t] |
| apply initial_prefix (n := t.length) (m := w.length) (by dsimp [n] at hlen; omega) |
| intro i hi |
| exact (show Z i = H (2*i+1) from by simpa using hH (2*i+1) (by dsimp [n]; omega)) |
| · intro A B hA hB |
| have ha := (realizes_initial _ _ _).mp hA |
| have hb := (realizes_initial _ _ _).mp hB |
| refine ⟨w,hwD,?_⟩ |
| apply Agree.mono (n := 2*w.length) _ (by omega) |
| apply agree_of_columns |
| · intro i hi |
| simpa [column,H] using ha i hi |
| · intro i hi |
| simpa [column,H] using hb i hi |
|
|
| def First (D : List Bool → Prop) (t s : List Bool) : Prop := |
| ∃ u, t <+: u ∧ Rectangle D s u |
|
|
| theorem first_denseOpen (D : List Bool → Prop) (hD : DenseOpen D) (t : List Bool) : |
| DenseOpen (First D t) := by |
| constructor |
| · intro s |
| obtain ⟨u,v,hs,ht,hr⟩ := rectangle_dense D hD s t |
| exact ⟨u,hs,v,ht,hr⟩ |
| · intro s u hsu |
| rintro ⟨v,ht,hr⟩ |
| exact ⟨v,ht,fun Y Z hY hZ => hr Y Z (realizes_mono hsu hY) hZ⟩ |
|
|
| def FirstFamily (D : ℕ → List Bool → Prop) (n : ℕ) : List Bool → Prop := |
| First (D (Nat.unpair n).1) (BorelGeneric.word (Nat.unpair n).2) |
|
|
| theorem firstFamily_denseOpen (D : ℕ → List Bool → Prop) (hD : ∀ n, DenseOpen (D n)) : |
| ∀ n, DenseOpen (FirstFamily D n) := |
| fun index => first_denseOpen _ (hD (Nat.unpair index).1) _ |
|
|
| def Fiber (D : ℕ → List Bool → Prop) (Y : Oracle) (n : ℕ) (t : List Bool) : Prop := |
| ∃ s, Realizes s Y ∧ Rectangle (D n) s t |
|
|
| theorem fiber_denseOpen (D : ℕ → List Bool → Prop) (Y : Oracle) |
| (hY : GenericFor (FirstFamily D) Y) (n : ℕ) : DenseOpen (Fiber D Y n) := by |
| constructor |
| · intro t |
| obtain ⟨k,hk⟩ := BorelGeneric.word_surjective t |
| obtain ⟨s,hs,hYs⟩ := hY (Nat.pair n k) |
| change First (D (Nat.unpair (Nat.pair n k)).1) |
| (BorelGeneric.word (Nat.unpair (Nat.pair n k)).2) s at hs |
| simp only [Nat.unpair_pair,hk] at hs |
| obtain ⟨u,htu,hr⟩ := hs |
| exact ⟨u,htu,s,hYs,hr⟩ |
| · intro t u htu |
| rintro ⟨s,hYs,hr⟩ |
| exact ⟨s,hYs,fun A B hA hB => hr A B hA (realizes_mono htu hB)⟩ |
|
|
| theorem fiber_isOpen (D : ℕ → List Bool → Prop) (n : ℕ) (t : List Bool) : |
| IsOpen {Y | Fiber D Y n t} := by |
| have he : {Y | Fiber D Y n t} = |
| ⋃ s : List Bool, ⋃ (_ : Rectangle (D n) s t), {Y | Realizes s Y} := by |
| ext Y |
| simp only [Fiber,mem_ofPred_eq,mem_iUnion] |
| aesop |
| rw [he] |
| exact isOpen_iUnion (fun s => isOpen_iUnion (fun _ => realizes_isOpen s)) |
|
|
| theorem join_generic (D : ℕ → List Bool → Prop) (Y Z : Oracle) |
| (hZ : GenericFor (Fiber D Y) Z) : GenericFor D (join Y Z) := by |
| intro n |
| obtain ⟨t,⟨s,hYs,hr⟩,hZt⟩ := hZ n |
| exact hr Y Z hYs hZt |
|
|
| theorem borel_fiber_selection (D : ℕ → List Bool → Prop) : |
| ∃ Z : {Y : Oracle // GenericFor (FirstFamily D) Y} → Oracle, Measurable Z ∧ |
| ∀ Y, GenericFor D (join Y.val (Z Y)) := by |
| let R := fun Y : {Y : Oracle // GenericFor (FirstFamily D) Y} => |
| fun n t => Fiber D Y.val n (BorelGeneric.word t) |
| have hd : ∀ Y n s, ∃ t, BorelGeneric.Ext s t ∧ R Y n t := by |
| intro Y n s |
| exact coded_dense _ (fiber_denseOpen D Y.val Y.property n) s |
| have hm : ∀ n t, MeasurableSet {Y | R Y n t} := by |
| intro n t |
| exact (fiber_isOpen D n (BorelGeneric.word t)).measurableSet.preimage measurable_subtype_coe |
| obtain ⟨Z,hZ,hreq⟩ := BorelGeneric.borel_generic_selection R hd hm |
| refine ⟨Z,hZ,fun Y => join_generic D Y.val (Z Y) ?_⟩ |
| intro n |
| obtain ⟨t,ht,hZt⟩ := hreq Y n |
| exact ⟨BorelGeneric.word t,ht,(realizes_iff_meets _ _).mpr hZt⟩ |
|
|
| end TuringRigidity.GenericFactor |
|
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| end OAI |
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