File size: 4,216 Bytes
b317718 | 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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | 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
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