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f1f7953 | 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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
namespace OAI
section
namespace Pureness
variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
theorem uniform_index_of_increasing_open_cover (U : ℕ → Set T)
(hopen : ∀ k, IsOpen (U k)) (hmono : Monotone U)
(hcover : ∀ τ, ∃ k, τ ∈ U k) : ∃ k, ∀ τ, τ ∈ U k := by
obtain ⟨k, hk⟩ := isCompact_univ.elim_directed_cover U hopen
(fun τ _ => Set.mem_iUnion.mpr (hcover τ)) hmono.directed_le
exact ⟨k, fun τ => hk (Set.mem_univ τ)⟩
theorem uniform_strict_index {R : Type*} [CompleteLinearOrder R]
(F : T → R) (g : ℕ → T → R)
(hopen : ∀ k, IsOpen {τ | F τ < g k τ})
(hmono : ∀ τ, Monotone (fun k => g k τ))
(hgap : ∀ τ, F τ < ⨆ k, g k τ) :
∃ k, ∀ τ, F τ < g k τ := by
apply uniform_index_of_increasing_open_cover (fun k => {τ | F τ < g k τ}) hopen
· intro i j hij τ hτ
exact lt_of_lt_of_le hτ (hmono τ hij)
· intro τ
exact lt_iSup_iff.mp (hgap τ)
end Pureness
end
end OAI
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