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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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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