| import OAI.NumberTheory.DukePrimeDegree.Entropy.FiniteFutureCodes |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.HeightObservationEntropy |
|
|
| namespace OAI |
|
|
| /-! # Finite codes converging to the future of a countable observation |
|
|
| Both the time interval and the clipping level increase. The limiting atom |
| records every future value of the original countable observation. |
| -/ |
|
|
| noncomputable section |
| open Set MeasureTheory |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| def countableFutureCode {Y : Type*} (T : Y → Y) (Q : Y → ℕ) (m : ℕ) |
| (y : Y) : Fin (Fintype.card (Fin m → Fin (m+1))) := |
| Fintype.equivFin _ (fun i : Fin m => heightClip Q m (T^[i.val+1] y)) |
|
|
| lemma countableFutureCode_measurable {Y : Type*} [MeasurableSpace Y] |
| {T : Y → Y} (hT : Measurable T) {Q : Y → ℕ} (hQ : Measurable Q) (m : ℕ) : |
| Measurable (countableFutureCode T Q m) := |
| (measurable_of_finite (Fintype.equivFin (Fin m → Fin (m+1)))).comp |
| (measurable_pi_iff.mpr fun i => (heightClip_measurable hQ m).comp (hT.iterate (i.val+1))) |
|
|
| def truncateCountableFutureCode {m n : ℕ} (hmn : m ≤ n) |
| (v : Fin (Fintype.card (Fin n → Fin (n+1)))) : |
| Fin (Fintype.card (Fin m → Fin (m+1))) := |
| Fintype.equivFin _ (fun i : Fin m => |
| ⟨min ((Fintype.equivFin (Fin n → Fin (n+1))).symm v |
| ⟨i.val,lt_of_lt_of_le i.isLt hmn⟩).val m, |
| Nat.lt_succ_of_le (min_le_right _ _)⟩) |
|
|
| lemma truncate_countableFutureCode {Y : Type*} {m n : ℕ} (T : Y → Y) (Q : Y → ℕ) |
| (hmn : m ≤ n) (y : Y) : |
| truncateCountableFutureCode hmn (countableFutureCode T Q n y) = |
| countableFutureCode T Q m y := by |
| simp only [truncateCountableFutureCode,countableFutureCode,Equiv.symm_apply_apply] |
| apply congrArg (Fintype.equivFin (Fin m → Fin (m+1))) |
| funext i |
| apply Fin.ext |
| change min (min (Q (T^[i.val+1] y)) n) m = min (Q (T^[i.val+1] y)) m |
| omega |
|
|
| lemma countableFutureCode_eq_of_longer {Y : Type*} {m n : ℕ} |
| (T : Y → Y) (Q : Y → ℕ) (hmn : m ≤ n) {x y : Y} |
| (h : countableFutureCode T Q n x=countableFutureCode T Q n y) : |
| countableFutureCode T Q m x=countableFutureCode T Q m y := by |
| have H := congrArg (truncateCountableFutureCode hmn) h |
| simpa only [truncate_countableFutureCode] using H |
|
|
| lemma all_countableFutureCode_eq_iff {Y : Type*} |
| (T : Y → Y) (Q : Y → ℕ) (x y : Y) : |
| (∀ m, countableFutureCode T Q m x=countableFutureCode T Q m y) ↔ |
| ∀ k : ℕ, Q (T^[k+1] x)=Q (T^[k+1] y) := by |
| constructor |
| · intro h k |
| let m := max k (max (Q (T^[k+1] x)) (Q (T^[k+1] y)))+1 |
| have hk : k < m := by dsimp only [m]; omega |
| have hx : Q (T^[k+1] x) < m := by dsimp only [m]; omega |
| have hy : Q (T^[k+1] y) < m := by dsimp only [m]; omega |
| have he := congrArg (fun v => |
| ((Fintype.equivFin (Fin m → Fin (m+1))).symm v ⟨k,hk⟩).val) (h m) |
| simp only [countableFutureCode,Equiv.symm_apply_apply] at he |
| change min (Q (T^[k+1] x)) m=min (Q (T^[k+1] y)) m at he |
| omega |
| · intro h m |
| apply congrArg (Fintype.equivFin (Fin m → Fin (m+1))) |
| funext i |
| apply Fin.ext |
| change min (Q (T^[i.val+1] x)) m=min (Q (T^[i.val+1] y)) m |
| rw [h i.val] |
|
|
| lemma transverse_countableFutureCode_monotone |
| {S V Y : Type*} [MeasurableSpace S] [MeasurableSpace V] |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) : |
| Monotone (fun m => transverseLabelSigma (fun z => countableFutureCode T Q m (F z))) := by |
| intro m n hmn A hA |
| obtain ⟨B,hB,rfl⟩ := hA |
| let f : S × Fin (Fintype.card (Fin n → Fin (n+1))) → |
| S × Fin (Fintype.card (Fin m → Fin (m+1))) := |
| fun z => (z.1,truncateCountableFutureCode hmn z.2) |
| have hf : Measurable f := measurable_fst.prodMk |
| ((measurable_of_finite (truncateCountableFutureCode hmn)).comp measurable_snd) |
| refine ⟨f ⁻¹' B,hf hB,?_⟩ |
| ext z |
| simp only [Set.mem_preimage,f,truncate_countableFutureCode] |
|
|
| lemma countableFutureCode_fibers_antitone {S V Y : Type*} |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) (z : S × V) : |
| Antitone (fun m => labelFiber (fun z => countableFutureCode T Q m (F z)) z) := by |
| intro m n hmn t ht |
| exact countableFutureCode_eq_of_longer T Q hmn ht |
|
|
| def countableFutureFiber {S V Y : Type*} (T : Y → Y) (Q : Y → ℕ) |
| (F : S × V → Y) (z : S × V) : Set V := |
| {t | ∀ k : ℕ, Q (T^[k+1] (F (z.1,t)))=Q (T^[k+1] (F z))} |
|
|
| lemma countableFutureFiber_eq_iInter {S V Y : Type*} |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) (z : S × V) : |
| countableFutureFiber T Q F z = |
| ⋂ m, labelFiber (fun z => countableFutureCode T Q m (F z)) z := by |
| ext t |
| simp only [countableFutureFiber,labelFiber,mem_iInter,mem_ofPred_eq] |
| exact (all_countableFutureCode_eq_iff T Q (F (z.1,t)) (F z)).symm |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
