File size: 2,378 Bytes
5711324 | 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 | import OAI.Analysis.CircleDomains.Topology.BasePathCodes
namespace OAI
/-! Forgetting auxiliary dust and tuple exclusions maps an allowed endpoint
code into the base-code family without changing its path or tags. -/
open Set MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
theorem tagTupleExcluded_mono
{S T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle)}
(h : ∀ n l, S n l ⊆ T n l) : tagTupleExcluded S ⊆ tagTupleExcluded T := by
apply iUnion_mono
intro n
apply iUnion_mono
intro l
exact image_mono (Set.prod_mono (h n l) Subset.rfl)
theorem tagRules_relax {X : Type*} [MetricSpace X]
{H : ℕ → Set (UnitAddCircle × X)}
{S T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle)}
(h : ∀ n l, S n l ⊆ T n l) {Γ : TaggedPath X} (hΓ : TagRules H T Γ) :
TagRules H S Γ := by
intro k
exact ⟨(hΓ k).1, (hΓ k).2.1, fun hk => (hΓ k).2.2 (tagTupleExcluded_mono h hk)⟩
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [MeasurableSpace C(PathTime, X)]
theorem allowedTaggedPath_forget (d : PathCodeData X Y) (ν : Measure X)
(T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle))
{Γ : TaggedPath X} (hΓ : AllowedTaggedPath d ν T Γ) :
AllowedTaggedPath d 0 (fun _ _ => ∅) Γ := by
refine ⟨hΓ.1, tagRules_relax (fun _ _ => empty_subset _) hΓ.2.1,
hΓ.2.2.1, hΓ.2.2.2.1, ?_⟩
simp
theorem endpointCode_forget (d : PathCodeData X Y) (ν : Measure X)
(T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle))
{Γ : TaggedPath X} (hΓ : EndpointCode d ν T Γ) :
EndpointCode d 0 (fun _ _ => ∅) Γ :=
⟨allowedTaggedPath_forget d ν T hΓ.1, hΓ.2⟩
def forgetCode (d : PathCodeData X Y) (ν : Measure X)
(T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle))
(Γ : {Γ : TaggedPath X // EndpointCode d ν T Γ}) : BasePathCode d :=
⟨Γ.val, endpointCode_forget d ν T Γ.property⟩
theorem measurable_forgetCode (d : PathCodeData X Y) (ν : Measure X)
(T : ∀ n : ℕ, (Fin (n + 1) → ℕ) → Set (Fin (n + 1) → UnitAddCircle)) :
Measurable (forgetCode d ν T) := measurable_subtype_coe.subtype_mk
end CircleDomainRigidity
end
end OAI
|