File size: 3,644 Bytes
dd14c26 | 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 | import OAI.Probability.LatticePassage.Stationarity
import OAI.Probability.LatticePassage.LocalPassage
namespace OAI
namespace CompanionFPP
open MeasureTheory
noncomputable instance encodable_lattice_walk (x y : Vertex) : Encodable (lattice.Walk x y) :=
Encodable.ofCountable _
noncomputable def enumerateWalk (x y : Vertex) (n : ℕ) : lattice.Walk x y :=
(Encodable.decode (α := lattice.Walk x y) n).getD (Classical.choice (lattice_walk x y))
lemma enumerateWalk_surjective (x y : Vertex) : Function.Surjective (enumerateWalk x y) := by
intro p
exact ⟨Encodable.encode p, by simp [enumerateWalk]⟩
def minimizesLocally (x y : Vertex) (R n : ℕ) (ω : Environment) : Prop :=
(enumerateWalk x y n).IsPath ∧ (enumerateWalk x y n).length < R ∧
pathCost ω (enumerateWalk x y n) = localPassageTime ω x y R
lemma exists_minimizesLocally (x y : Vertex) (R : ℕ) (hR : latticeDistance x y < R)
(ω : Environment) : ∃ n, minimizesLocally x y R n ω := by
obtain ⟨p,hp,hl,hm⟩ := local_geodesic_exists ω x y R hR
obtain ⟨n,rfl⟩ := enumerateWalk_surjective x y p
exact ⟨n,hp,hl,hm⟩
lemma measurableSet_minimizesLocally (x y : Vertex) (R n : ℕ) :
MeasurableSet {ω : Environment | minimizesLocally x y R n ω} := by
unfold minimizesLocally
by_cases hp : (enumerateWalk x y n).IsPath
· by_cases hl : (enumerateWalk x y n).length < R
· simpa [hp,hl] using measurableSet_eq_fun (measurable_pathCost (enumerateWalk x y n))
(measurable_localPassageTime x y R)
· simp [hl]
· simp [hp]
noncomputable def localSelectionIndex (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (ω : Environment) : ℕ := by
classical
exact Nat.find (exists_minimizesLocally x y R hR ω)
@[fun_prop] theorem measurable_localSelectionIndex (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) : Measurable (localSelectionIndex x y R hR) := by
classical
exact measurable_find _ (measurableSet_minimizesLocally x y R)
noncomputable def selectedLocalPath (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (ω : Environment) : lattice.Walk x y :=
enumerateWalk x y (localSelectionIndex x y R hR ω)
theorem selectedLocalPath_spec (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (ω : Environment) :
(selectedLocalPath x y R hR ω).IsPath ∧
(selectedLocalPath x y R hR ω).length < R ∧
pathCost ω (selectedLocalPath x y R hR ω) = localPassageTime ω x y R := by
classical
exact Nat.find_spec (exists_minimizesLocally x y R hR ω)
theorem measurable_selectedLocalPath_observation (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (f : lattice.Walk x y → ℝ) :
Measurable (fun ω => f (selectedLocalPath x y R hR ω)) := by
exact (measurable_of_countable (f := fun n : ℕ => f (enumerateWalk x y n))).comp
(measurable_localSelectionIndex x y R hR)
theorem selectedLocalPath_eq_of_agree (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (ω ω' : Environment)
(h : ∀ e ∈ localEdges x R, ω e = ω' e) :
selectedLocalPath x y R hR ω = selectedLocalPath x y R hR ω' := by
classical
have he : ∀ n, minimizesLocally x y R n ω ↔ minimizesLocally x y R n ω' := by
intro n
unfold minimizesLocally
by_cases hp : (enumerateWalk x y n).IsPath
· by_cases hl : (enumerateWalk x y n).length < R
· rw [local_pathCost_eq ω ω' h _ hp hl,
localPassageTime_eq_of_agree ω ω' x y R h]
· simp [hl]
· simp [hp]
unfold selectedLocalPath localSelectionIndex
congr 1
exact Nat.find_congr' (fun {n} => he n)
end CompanionFPP
end OAI
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