openAI-math-dataset / lean /OAI /Analysis /BiholderTransport /LinearAlgebra /CompactLocalPositiveBounds.lean
| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Regularity.ContDiffParametricQuadratic | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace WeakMTWTransport | |
| lemma compact_local_positive_bounds {A : Type*} [TopologicalSpace A] | |
| {K : Set A} (hK : IsCompact K) {P : ℝ → ℝ → A → Prop} | |
| (hmono : ∀ {r R b B : ℝ}, 0<r → r≤R → b≤B → ∀ a, P R b a → P r B a) | |
| (hlocal : ∀ a∈K, ∃ r b : ℝ, 0<r ∧ 0<b ∧ ∀ᶠ z in 𝓝 a, P r b z) : | |
| ∃ r b : ℝ, 0<r ∧ 0<b ∧ ∀ a∈K, P r b a := by | |
| obtain ⟨r,hr,b,hb,H⟩ := compact_eventually_uniform_radius_bound | |
| (T := Unit) (t := ()) (P := fun r b (_ : Unit) a => P r b a) hK | |
| (fun hr hR hB _ a ha => hmono hr hR hB a ha) (by | |
| intro a ha | |
| obtain ⟨r,b,hr,hb,H⟩ := hlocal a ha | |
| exact ⟨r,hr,b,hb,continuousAt_snd.eventually H⟩) | |
| exact ⟨r,b,hr,hb,H.self_of_nhds⟩ | |
| end WeakMTWTransport | |
| end | |
| end | |
| end | |
| end OAI | |