| 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 |
|
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| end OAI |
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|
