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a2ee19a | 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import OAI.Probability.RadialTransition.MultiConvolution
namespace OAI
universe uX
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
attribute [local instance] Classical.propDecidable
lemma configurationMultiset_fibers {X : Type uX} {ι I : Type} [Fintype ι] [Fintype I]
(x : ι → X) (f : ι → I) :
∑ q : I, ((Finset.univ.filter (fun j => f j = q)).val.map x) = configurationMultiset x := by
classical
have hq (q : I) : ((Finset.univ.filter (fun j => f j=q)).val.map x) =
∑ j : ι, if f j=q then ({x j}:Multiset X) else 0 := by
rw [← Finset.sum_filter]
change _ = ((Finset.univ.filter (fun j => f j=q)).val.map (fun j => ({x j}:Multiset X))).sum
simpa only [Multiset.map_map, Function.comp_def] using
(Multiset.sum_map_singleton ((Finset.univ.filter (fun j => f j=q)).val.map x)).symm
simp_rw [hq]
rw [Finset.sum_comm]
simp [configurationMultiset]
section Embed
variable {X : Type uX} {I Q : Type} [Fintype I] [Fintype Q]
lemma indexedClusterProduct_embedding (W : I → Multiset X → ℝ≥0∞)
(h0 : ∀ i, 1 ≤ W i 0) (f : Q → I) (hf : Function.Injective f) (s : Q → Multiset X) :
(∏ q, W (f q) (s q)) ≤ indexedClusterProduct W (∑ q, s q) := by
classical
let u (i : I) : Multiset X := ∑ q : Q, if f q=i then s q else 0
have huv (q : Q) : u (f q) = s q := by
simp [u, hf.eq_iff]
have huz (i : I) (hi : i ∉ Finset.univ.image f) : u i = 0 := by
apply Finset.sum_eq_zero
intro q _
exact ite_eq_right (fun he => hi (Finset.mem_image.mpr ⟨q,Finset.mem_univ _,he⟩))
have hsum : ∑ i, u i = ∑ q, s q := by
simp only [u]
rw [Finset.sum_comm]
simp
have hp : (∏ q, W (f q) (s q)) = ∏ i ∈ Finset.univ.image f, W i (u i) := by
rw [Finset.prod_image (fun a _ b _ hab => hf hab)]
simp_rw [huv]
rw [hp]
have hle : (∏ i ∈ Finset.univ.image f, W i (u i)) ≤ ∏ i, W i (u i) := by
apply Finset.prod_le_prod_of_subset_of_one_le₀ (Finset.subset_univ _) (fun _ _ => bot_le)
intro i _ hi
rw [huz i hi]
exact h0 i
apply hle.trans
rw [← hsum]
exact indexedClusterProduct_ge W u
end Embed
end RadialTransition
end
end OAI
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