| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.AuxiliaryChecks |
|
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {ι C : Type*} [Fintype ι] [Fintype C] [DecidableEq ι] |
| {X : ι → Type*} [∀i,MeasurableSpace (X i)] |
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| omit [Fintype ι] in |
| lemma uniformInsertionOn_of_check_failures (checks : C → LocalCheck X) |
| (μ : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (μ i)] |
| (p : ι → C → ℝ≥0∞) (a : ℝ≥0∞) (r : Finset ι) |
| (hfail : ∀i∈r, ∀c, i∈(checks c).support → ∀x : ∀j,X j, |
| μ i {y | ¬(checks c).pass (Function.update x i y)} ≤ p i c) |
| (htotal : ∀i∈r,a+(∑c∈Finset.univ.filter (fun c ↦ i∈(checks c).support),p i c) ≤ 1) : |
| UniformInsertionOn checks μ a r := by |
| classical |
| intro i hir s hi x |
| let B : Set (X i) := {y | Function.update x i y ∈ insertionEvent checks s i} |
| have hm : Measurable (fun y : X i ↦ Function.update x i y) := by |
| fun_prop |
| have hB : MeasurableSet B := (measurableSet_insertionEvent checks s i).preimage hm |
| have hsub : Bᶜ ⊆ ⋃c∈Finset.univ.filter (fun c ↦ i∈(checks c).support), |
| {y | ¬(checks c).pass (Function.update x i y)} := by |
| intro y hy |
| classical |
| simp only [B,Set.mem_compl_iff,Set.mem_ofPred_eq,insertionEvent] at hy |
| push Not at hy |
| rcases hy with ⟨c,hc,hcs,hp⟩ |
| have hci : i∈(checks c).support := by |
| by_contra hn |
| apply hcs |
| intro j hj |
| have h := hc hj |
| rcases Finset.mem_insert.mp h with he|hs |
| · subst j; contradiction |
| · exact hs |
| exact mem_iUnion.2 ⟨c,mem_iUnion.2 ⟨by simp [hci],hp⟩⟩ |
| have hb : μ i Bᶜ ≤ ∑c∈Finset.univ.filter (fun c ↦ i∈(checks c).support),p i c := by |
| apply (measure_mono hsub).trans |
| apply (measure_biUnion_finset_le _ _).trans |
| exact Finset.sum_le_sum fun c hc ↦ hfail i hir c (Finset.mem_filter.mp hc).2 x |
| have hadd : a+μ i Bᶜ ≤ μ i B+μ i Bᶜ := by |
| rw [measure_add_measure_compl hB] |
| exact (add_le_add_right hb a).trans (by simpa using htotal i hir) |
| have ha : a≤μ i B := (ENNReal.add_le_add_iff_right (measure_ne_top _ _)).mp hadd |
| have he : (fun y ↦ (insertionEvent checks s i).indicator (fun _ ↦ (1 : ℝ≥0∞)) (Function.update x i y))= |
| B.indicator (fun _ ↦ (1 : ℝ≥0∞)) := by funext y; simp [B,Set.indicator_apply] |
| rw [he,lintegral_indicator hB] |
| simpa using ha |
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| end HardSphere.Comparison |
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| end |
| end |
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| end |
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| end OAI |
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