| import Mathlib |
| import OAI.Geometry.Cannon.Flows.Domain |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set MeasureTheory |
| open scoped BigOperators |
| open scoped NNReal ENNReal |
| open Set Metric |
| open Set Filter Topology |
| open Set Filter Topology Metric |
| open Set Metric Topology |
| open Set unitInterval |
|
|
| namespace CannonMagnification |
|
|
| open Set Metric Filter Topology |
|
|
| lemma continuousOn_onDomain {X : Type*} [TopologicalSpace X] {Ω : Set X} |
| (f : C(Ω,ℝ)) : ContinuousOn (onDomain f) Ω := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| convert f.continuous using 1 |
| ext x |
| exact onDomain_eq f x.val x.property |
|
|
| lemma uniform_onDomain {X : Type*} [MetricSpace X] {Ω : Set X} |
| (f : ℕ → X → ℝ) (F : C(Ω,ℝ)) |
| (hlim : ∀ K : Set Ω, IsCompact K → |
| TendstoUniformlyOn (fun i z => f i z.val) F atTop K) |
| (K : Set X) (hK : IsCompact K) (hKΩ : K ⊆ Ω) : |
| TendstoUniformlyOn f (onDomain F) atTop K := by |
| let K' : Set Ω := Subtype.val ⁻¹' K |
| have hK' : IsCompact K' := IsEmbedding.subtypeVal.isInducing.isCompact_preimage' hK |
| (by simpa only [Subtype.range_coe] using hKΩ) |
| intro u hu |
| filter_upwards [hlim K' hK' u hu] with i hi |
| intro z hz |
| simpa only [onDomain_eq F z (hKΩ hz)] using hi ⟨z,hKΩ hz⟩ hz |
|
|
| end CannonMagnification |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
| |
