| 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 | |