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92416a2 | 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 OAI.Dynamics.StandardMap.CriticalLaws
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open Set Filter
open scoped Topology Classical
namespace StandardMapEntropy
lemma compact_nonaffine_finite_tests (F : Set DistanceArray) (hF : IsCompact F) (hFa : F⊆affineLocusᶜ) :
∃ t : Finset ArrayTestIndex, ∃ c : ℝ, 0<c ∧ ∀ d∈F,c ≤ ∑ i∈t,arrayTest i d := by
classical
by_cases hne : F.Nonempty
· let U : ArrayTestIndex → Set DistanceArray := fun i => {d | 0<arrayTest i d}
have hU : ∀ i,IsOpen (U i) := fun i => isOpen_lt continuous_const (continuous_arrayTest i)
have hcover : F⊆⋃ i,U i := by
intro d hd
obtain ⟨i,hi⟩ := (nonaffine_iff_positive_test d).mp (hFa hd)
exact Set.mem_iUnion.mpr ⟨i,hi⟩
obtain ⟨t,ht⟩ := hF.elim_finite_subcover U hU hcover
let f : DistanceArray → ℝ := fun d => ∑ i∈t,arrayTest i d
have hcont : Continuous f := continuous_finsetSum _ (fun i hi => continuous_arrayTest i)
have hpos : ∀ d∈F,0<f d := by
intro d hd
obtain ⟨i,hi,hd⟩ := Set.mem_iUnion₂.mp (ht hd)
exact lt_of_lt_of_le hd (Finset.single_le_sum (fun j hj => arrayTest_nonneg j d) hi)
obtain ⟨d,hd,hmin⟩ := hF.exists_isMinOn hne hcont.continuousOn
exact ⟨t,f d,hpos d hd,fun e he => hmin he⟩
· exact ⟨∅,1,by norm_num,fun d hd => (hne ⟨d,hd⟩).elim⟩
end StandardMapEntropy
end OAI
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