| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.Realization |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
|
|
| namespace RepeatedLocalization |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
|
|
| lemma through_init {N t : ℕ} (w : History N (t+1)) (i : Fin t) : |
| through (Fin.init w) i = through w i.castSucc := by |
| funext j |
| rfl |
|
|
| noncomputable def extendRealization {n N t : ℕ} {s s' : AnalysisState n} |
| {f : s.Index → History N t} {q : s.Index → FiniteChannel (Coordinate n) (Fin N)} |
| {g : s'.Index → s.Index × Fin N} (R : Realization s f) (E : ExtensionWitness s q s' g) : |
| Realization s' (fun j => Fin.snoc (α := fun _ : Fin (t+1) => Fin N) (f (g j).1) (g j).2) := by |
| classical |
| let rd := fun (z : Coordinate n) (w : History N (t+1)) => E.choose (R.read z (Fin.init w),w (Fin.last t)) z |
| have hrd : Measurable (fun zw : Coordinate n × History N (t+1) => rd zw.1 zw.2) := by |
| apply E.choose_meas.comp (f := fun zw : Coordinate n × History N (t+1) => ((R.read zw.1 (Fin.init zw.2),zw.2 (Fin.last t)),zw.1)) |
| exact ((R.read_meas.comp (measurable_fst.prodMk ((measurable_of_countable (fun w : History N (t+1) => Fin.init w)).comp measurable_snd))).prodMk |
| ((measurable_of_countable (fun w : History N (t+1) => w (Fin.last t))).comp measurable_snd)).prodMk measurable_fst |
| have herase (z : Coordinate n) (w : History N (t+1)) : |
| g (rd z w) = (R.read z (Fin.init w),w (Fin.last t)) := E.erase_choose _ _ |
| let Q : Disclosure n N (t+1) := Fin.lastCases (fun z w => s'.cell (rd z w)) R.Q |
| let tr : s'.Index → Fin (t+1) → Cell n := fun j => Fin.snoc (R.trace (g j).1) (s'.cell j) |
| refine { |
| Q := Q |
| Qmeas := ?_ |
| Qfinite := ?_ |
| read := rd |
| read_meas := hrd |
| erase_read := ?_ |
| chosen := ?_ |
| trace := tr |
| trace_read := ?_ |
| trace_cube := ?_ |
| trace_below := ?_ |
| trace_nested := ?_ |
| cell_cube := fun j => (E.subset j).trans (R.cell_cube (g j).1) |
| read_mem := ?_ |
| depth_read := ?_ } |
| · intro i |
| refine Fin.lastCases ?_ (fun j => ?_) i |
| · simp only [Q,Fin.lastCases_last] |
| change Measurable (fun zw : Coordinate n × History N (t+1) => s'.cell (rd zw.1 zw.2)) |
| exact (measurable_of_countable s'.cell).comp hrd |
| · simp only [Q,Fin.lastCases_castSucc] |
| change Measurable (fun zw : Coordinate n × History N (j.val+1) => R.Q j zw.1 zw.2) |
| exact R.Qmeas j |
| · intro i |
| refine Fin.lastCases ?_ (fun j => ?_) i |
| · simp only [Q,Fin.lastCases_last] |
| change (Set.range (fun zw : Coordinate n × History N (t+1) => s'.cell (rd zw.1 zw.2))).Finite |
| apply (Set.finite_range s'.cell).subset |
| rintro c ⟨zw,rfl⟩ |
| exact ⟨rd zw.1 zw.2,rfl⟩ |
| · simp only [Q,Fin.lastCases_castSucc] |
| change (Set.range (fun zh : Coordinate n × History N (j.val+1) => R.Q j zh.1 zh.2)).Finite |
| exact R.Qfinite j |
| · intro z w |
| rw [herase,R.erase_read] |
| exact Fin.snoc_init_self w |
| · intro j z hz |
| dsimp only [rd] |
| rw [Fin.init_snoc,Fin.snoc_last,R.chosen (g j).1 z (E.positive_parent j z hz)] |
| exact E.chosen j z hz |
| · intro z w i |
| refine Fin.lastCases ?_ (fun j => ?_) i |
| · simp [tr,Q,through_last] |
| · simp only [tr,Q,Fin.snoc_castSucc,Fin.lastCases_castSucc] |
| rw [herase] |
| simpa only [through_init] using R.trace_read z (Fin.init w) j |
| · intro j i |
| refine Fin.lastCases ?_ (fun k => ?_) i |
| · simp only [tr,Fin.snoc_last] |
| exact (E.subset j).trans (R.cell_cube _) |
| · simp only [tr,Fin.snoc_castSucc] |
| exact R.trace_cube _ k |
| · intro j i |
| refine Fin.lastCases ?_ (fun k => ?_) i |
| · simp only [tr,Fin.snoc_last] |
| exact Subset.rfl |
| · simp only [tr,Fin.snoc_castSucc] |
| exact (E.subset j).trans (R.trace_below _ k) |
| · intro k i j |
| refine Fin.lastCases ?_ (fun a => ?_) i |
| · refine Fin.lastCases ?_ (fun b => ?_) j |
| · intro _; exact Subset.rfl |
| · intro h; exact False.elim (by have := b.isLt; change t ≤ b.val at h; omega) |
| · refine Fin.lastCases ?_ (fun b => ?_) j |
| · intro _ |
| simp only [tr,Fin.snoc_last,Fin.snoc_castSucc] |
| exact (E.subset k).trans (R.trace_below _ a) |
| · intro h |
| simpa only [tr,Fin.snoc_castSucc] using R.trace_nested (g k).1 a b h |
| · intro z hz w |
| exact E.contains _ z (R.read_mem z hz (Fin.init w)) |
| · intro z w |
| simp only [finalLevel,show 0 < t+1 by omega,dite_eq_left] |
| change (Q (Fin.last t) z (through w (Fin.last t))).1 = (s'.cell (rd z w)).1 |
| rw [through_last] |
| simp only [Q,Fin.lastCases_last] |
| |
| end RepeatedLocalization |
| |
| end OAI |
| |
