| import OAI.Dynamics.StandardMap.CriticalLaws |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| 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 |
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| end OAI |
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