File size: 3,136 Bytes
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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import OAI.Probability.RadialTransition.GrandBounds
namespace OAI
universe uX uY
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
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]
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)
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)
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
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⟩
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
end RadialTransition
end
end OAI
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