| import OAI.Probability.RadialTransition.GrandBounds |
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| namespace OAI |
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| universe uX uY |
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| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
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| lemma antidiagonal_map {X : Type uX} {Y : Type uY} (f : X → Y) (s : Multiset X) : |
| (s.map f).antidiagonal = s.antidiagonal.map (fun p => (p.1.map f, p.2.map f)) := by |
| induction s using Multiset.induction_on with |
| | empty => simp |
| | cons a s ih => |
| simp only [Multiset.map_cons, Multiset.antidiagonal_cons, ih, Multiset.map_add, Multiset.map_map] |
| congr 1 <;> apply Multiset.map_congr rfl <;> intro p hp <;> simp |
| |
| lemma powerset_val_map {X : Type uX} (s : Finset X) : |
| (s.powerset.val.map Finset.val) = s.val.powerset := by |
| simp [Finset.powerset, Multiset.map_pmap, Multiset.pmap_eq_map] |
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| lemma configurationMultiset_val {X : Type uX} {ι : Type} [Fintype ι] (x : ι → X) : |
| configurationMultiset x = (Finset.univ.val.map x) := by |
| simp only [configurationMultiset, Finset.sum] |
| simpa only [Multiset.map_map, Function.comp_def] using |
| Multiset.sum_map_singleton (Finset.univ.val.map x) |
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| lemma configurationMultiset_subtype {X : Type uX} {ι : Type} [Fintype ι] |
| (x : ι → X) (T : Finset ι) : |
| configurationMultiset (fun i : T => x i) = T.val.map x := by |
| unfold configurationMultiset |
| calc |
| (∑ i : T, {x i}) = ∑ i ∈ T, {x i} := Finset.sum_finset_coe (fun i => {x i}) T |
| _ = _ := by |
| change (T.val.map (fun i => {x i})).sum = T.val.map x |
| simpa only [Multiset.map_map, Function.comp_def] using Multiset.sum_map_singleton (T.val.map x) |
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| def clusterConvolution {X : Type uX} (F G : Multiset X → ℝ≥0∞) (s : Multiset X) : ℝ≥0∞ := |
| (s.antidiagonal.map (fun p => F p.1 * G p.2)).sum |
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| lemma clusterConvolution_add_ge {X : Type uX} (F G : Multiset X → ℝ≥0∞) (s u : Multiset X) : |
| F s*G u ≤ clusterConvolution F G (s+u) := by |
| apply Multiset.single_le_sum (fun _ _ => bot_le) |
| exact Multiset.mem_map.mpr ⟨(s,u), Multiset.mem_antidiagonal.mpr rfl, rfl⟩ |
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| lemma clusterConvolution_configuration {X : Type uX} {ι : Type} [Fintype ι] [DecidableEq ι] |
| (F G : Multiset X → ℝ≥0∞) (x : ι → X) : |
| clusterConvolution F G (configurationMultiset x) = |
| ∑ T : Finset ι, F (configurationMultiset (fun i : {i // i ∉ T} => x i)) * |
| G (configurationMultiset (fun i : T => x i)) := by |
| classical |
| rw [clusterConvolution, configurationMultiset_val, antidiagonal_map, |
| Multiset.antidiagonal_eq_map_powerset, ← powerset_val_map] |
| simp only [Multiset.map_map, Function.comp_def] |
| rw [Finset.powerset_univ] |
| apply Finset.sum_congr rfl |
| intro T _ |
| congr 1 |
| · have hval : Finset.univ.val-T.val = Tᶜ.val := by |
| change (Finset.univ \ T).val = Tᶜ.val |
| rfl |
| rw [hval] |
| congr 1 |
| let e : {i // i ∉ T} ≃ {i // i ∈ Tᶜ} := Equiv.subtypeEquivProp (by simp) |
| have hh := configurationMultiset_equiv e (fun i : {i // i ∈ Tᶜ} => x i) |
| rw [configurationMultiset_subtype] at hh |
| exact hh.symm |
| · congr 1 |
| exact (configurationMultiset_subtype x T).symm |
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| end RadialTransition |
| end |
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| end OAI |
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