| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.Refinement |
|
|
| 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_snoc_castSucc {N t : ℕ} (w : History N t) (v : Fin N) (i : Fin t) : |
| through (Fin.snoc (α := fun _ : Fin (t+1) => Fin N) w v) i.castSucc = through w i := by |
| funext j |
| change Fin.snoc (α := fun _ : Fin (t+1) => Fin N) w v |
| (Fin.castSucc ⟨j.val,Nat.lt_of_lt_of_le j.isLt i.isLt⟩) = _ |
| exact Fin.snoc_castSucc (α := fun _ : Fin (t+1) => Fin N) v w ⟨j.val,Nat.lt_of_lt_of_le j.isLt i.isLt⟩ |
|
|
| lemma through_last {N t : ℕ} (w : History N (t+1)) : through w (Fin.last t) = w := by |
| funext j |
| rfl |
|
|
| instance AnalysisState.indexMeasurableSpace {n : ℕ} (s : AnalysisState n) : MeasurableSpace s.Index := ⊤ |
|
|
| instance AnalysisState.indexMeasurableSingleton {n : ℕ} (s : AnalysisState n) : MeasurableSingletonClass s.Index := ⟨fun _ => trivial⟩ |
|
|
| structure Realization {n N t : ℕ} (s : AnalysisState n) (f : s.Index → History N t) where |
| Q : Disclosure n N t |
| Qmeas : ∀ i, Measurable (fun zh : Coordinate n × History N (i.val+1) => Q i zh.1 zh.2) |
| Qfinite : ∀ i, (Set.range (fun zh : Coordinate n × History N (i.val+1) => Q i zh.1 zh.2)).Finite |
| read : Coordinate n → History N t → s.Index |
| read_meas : Measurable (fun zw : Coordinate n × History N t => read zw.1 zw.2) |
| erase_read : ∀ z w, f (read z w) = w |
| chosen : ∀ j z, 0 < s.channel.prob j z → read z (f j) = j |
| trace : s.Index → Fin t → Cell n |
| trace_read : ∀ z w i, trace (read z w) i = Q i z (through w i) |
| trace_cube : ∀ j i, cellSet (trace j i) ⊆ initialCube n |
| trace_below : ∀ j i, cellSet (s.cell j) ⊆ cellSet (trace j i) |
| trace_nested : ∀ k (i j : Fin t), i ≤ j → cellSet (trace k j) ⊆ cellSet (trace k i) |
| cell_cube : ∀ j, cellSet (s.cell j) ⊆ initialCube n |
| read_mem : ∀ z ∈ initialCube n, ∀ w, z ∈ cellSet (s.cell (read z w)) |
| depth_read : ∀ z w, finalLevel Q (z,w) = (s.cell (read z w)).1 |
|
|
| lemma Realization.valid {n N t : ℕ} {s : AnalysisState n} {f : s.Index → History N t} |
| (R : Realization s f) : ValidDisclosure R.Q := by |
| refine ⟨R.Qmeas,R.Qfinite,?_,?_⟩ |
| · intro z hz w i |
| rw [← R.trace_read] |
| exact ⟨R.trace_below (R.read z w) i (R.read_mem z hz w),R.trace_cube _ i⟩ |
| · intro z hz w i j hij |
| rw [← R.trace_read,← R.trace_read] |
| exact R.trace_nested _ i j hij |
|
|
| noncomputable def initialRealization {n N : ℕ} (hn : 0 < n) : |
| Realization (initialState hn) (fun _ => (Fin.elim0 : History N 0)) where |
| Q := fun i => Fin.elim0 i |
| Qmeas := fun i => Fin.elim0 i |
| Qfinite := fun i => Fin.elim0 i |
| read := fun _ _ => () |
| read_meas := measurable_const |
| erase_read := fun _ _ => Subsingleton.elim _ _ |
| chosen := fun j _ _ => by change () = (j : Unit); exact Subsingleton.elim _ _ |
| trace := fun _ i => Fin.elim0 i |
| trace_read := fun _ _ i => Fin.elim0 i |
| trace_cube := fun _ i => Fin.elim0 i |
| trace_below := fun _ i => Fin.elim0 i |
| trace_nested := fun _ i => Fin.elim0 i |
| cell_cube := fun _ => by change cellSet (rootCell n) ⊆ initialCube n; rw [cellSet_root] |
| read_mem := fun z hz _ => by change z ∈ cellSet (rootCell n); rwa [cellSet_root] |
| depth_read := fun _ _ => by simp [finalLevel,initialState,rootCell] |
|
|
| lemma ExtensionWitness.choose_meas {n N : ℕ} {s s' : AnalysisState n} |
| {q : s.Index → FiniteChannel (Coordinate n) (Fin N)} {g : s'.Index → s.Index × Fin N} |
| (E : ExtensionWitness s q s' g) : Measurable (fun iz : (s.Index × Fin N) × Coordinate n => E.choose iz.1 iz.2) := by |
| apply measurable_from_prod_countable_right |
| intro i |
| apply measurable_to_countable |
| intro z |
| exact E.choose_fiber i (E.choose i z) |
| |
| end RepeatedLocalization |
| |
| end OAI |
| |
