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110f4f3 | 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 | 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
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