| import OAI.Probability.RadialTransition.ConvolutionFock |
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| namespace OAI |
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| universe uX uM uY uZ |
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| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
| attribute [local instance] Classical.propDecidable |
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| lemma configurationMultiset_card {X : Type uX} {ι : Type} [Fintype ι] (x : ι → X) : |
| (configurationMultiset x).card = Fintype.card ι := by |
| rw [configurationMultiset_val, Multiset.card_map] |
| exact Finset.card_univ |
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| lemma configurationMultiset_sum_map {X : Type uX} {M : Type uM} [AddCommMonoid M] {ι : Type} [Fintype ι] |
| (x : ι → X) (f : X → M) : ((configurationMultiset x).map f).sum = ∑ i, f (x i) := by |
| rw [configurationMultiset_val, Multiset.map_map] |
| rfl |
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| lemma configurationMultiset_prod_map {X : Type uX} {M : Type uM} [CommMonoid M] {ι : Type} [Fintype ι] |
| (x : ι → X) (f : X → M) : ((configurationMultiset x).map f).prod = ∏ i, f (x i) := by |
| rw [configurationMultiset_val, Multiset.map_map] |
| rfl |
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| def nearCube (r : ℝ) (x : Position) : Set Position := |
| {y | ∀ k, |y k-x k| ≤ r} |
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| def nearMask (r : ℝ) (x y : Position) : ℝ≥0∞ := (nearCube r x).indicator (fun _ => 1) y |
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| def regionMask (A : Set Position) (x : Position) : ℝ≥0∞ := A.indicator (fun _ => 1) x |
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| def starWeight (A : Set Position) (r : ℝ) (s : Multiset Position) : ℝ≥0∞ := |
| (s.map (fun x => regionMask A x * (s.map (nearMask r x)).prod)).sum |
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| def emptyWeight {X : Type uX} (s : Multiset X) : ℝ≥0∞ := if s.card = 0 then 1 else 0 |
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| def cellClusterWeight (t α r : ℝ) (k : Cell) (s : Multiset Position) : ℝ≥0∞ := |
| emptyWeight s + if s.card ≤ 4 then ENNReal.ofReal (Real.exp (-t*α))*starWeight (gridRegion k) r s else 0 |
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| lemma measurable_regionMask {A : Set Position} (hA : MeasurableSet A) : Measurable (regionMask A) := |
| measurable_const.indicator hA |
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| lemma measurable_nearMask (r : ℝ) : Measurable (Function.uncurry (nearMask r)) := by |
| have hs : MeasurableSet {p : Position × Position | ∀ k, |p.2 k-p.1 k| ≤ r} := by |
| simp only [Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro k |
| exact measurableSet_le (((PiLp.continuous_apply 2 (fun _ : Fin 3 => ℝ) k).comp continuous_snd).sub |
| ((PiLp.continuous_apply 2 (fun _ : Fin 3 => ℝ) k).comp continuous_fst)).abs.measurable measurable_const |
| exact measurable_const.indicator hs |
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| lemma measurable_multiset_prod {Y : Type uY} {Z : Type uZ} [MeasurableSpace Y] |
| (s : Multiset Z) (f : Z → Y → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) : |
| Measurable (fun x => (s.map (fun i => f i x)).prod) := by |
| induction s using Multiset.induction_on with |
| | empty => simp |
| | cons a s ih => |
| simp only [Multiset.map_cons, Multiset.prod_cons] |
| exact measurable_mul.comp ((hf a).prodMk ih) |
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| lemma measurable_starWeight {A : Set Position} (hA : MeasurableSet A) (r : ℝ) : |
| MeasurableConfiguration (starWeight A r) := by |
| intro s ι _ |
| classical |
| have hprod (a : (ι → Position) → Position) (ha : Measurable a) : |
| Measurable (fun y => ((s+configurationMultiset y).map (nearMask r (a y))).prod) := by |
| simp only [Multiset.map_add, Multiset.prod_add, configurationMultiset_prod_map] |
| apply Measurable.mul |
| · apply measurable_multiset_prod |
| intro v |
| exact (measurable_nearMask r).comp (ha.prodMk measurable_const) |
| · apply Finset.measurable_prod |
| intro i _ |
| exact (measurable_nearMask r).comp (ha.prodMk (measurable_pi_apply i)) |
| have he (x : ι → Position) : starWeight A r (s+configurationMultiset x) = |
| (s.map (fun a => regionMask A a * ((s+configurationMultiset x).map (nearMask r a)).prod)).sum + |
| ∑ i, regionMask A (x i) * ((s+configurationMultiset x).map (nearMask r (x i))).prod := by |
| unfold starWeight |
| rw [Multiset.map_add, Multiset.sum_add, configurationMultiset_sum_map] |
| simp_rw [he] |
| apply Measurable.add |
| · apply measurable_multiset_sum |
| intro a |
| exact measurable_const.mul (hprod (fun _ => a) measurable_const) |
| · apply Finset.measurable_sum |
| intro i _ |
| exact ((measurable_regionMask hA).comp (measurable_pi_apply i)).mul |
| (hprod (fun x => x i) (measurable_pi_apply i)) |
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| lemma measurable_emptyWeight {X : Type uX} [MeasurableSpace X] : |
| MeasurableConfiguration (emptyWeight (X := X)) := by |
| intro s ι _ |
| simp only [emptyWeight, Multiset.card_add, configurationMultiset_card] |
| exact measurable_const |
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| lemma measurable_cellClusterWeight (t α r : ℝ) (k : Cell) : |
| MeasurableConfiguration (cellClusterWeight t α r k) := by |
| intro s ι _ |
| simp only [cellClusterWeight, Multiset.card_add, configurationMultiset_card] |
| split_ifs |
| · exact (measurable_emptyWeight s ι).add (measurable_const.mul (measurable_starWeight (measurableSet_gridRegion k) r s ι)) |
| · exact (measurable_emptyWeight s ι).add measurable_const |
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| end RadialTransition |
| end |
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| end OAI |
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