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9b6e014 | 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Probability.ContinuumTransition.RootVolume
namespace OAI
section
open scoped BigOperators ENNReal
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
lemma measurableSet_freeConfig {ι : Type} [Fintype ι] (t : ℝ) :
MeasurableSet {x : ι → Space | FreeConfig t x} := by
apply measurableSet_forall_labels
intro i
apply measurableSet_forall_labels
intro j
by_cases hij : i = j
· simp only [hij, ne_eq, not_true_eq_false, false_implies, ofPred_true, MeasurableSet.univ]
have hm : Measurable (fun x : ι → Space => ‖x i-x j‖) :=
((measurable_pi_apply i).sub (measurable_pi_apply j)).norm
have he : {x : ι → Space | i ≠ j → ‖x i-x j‖ < 1 →
Real.exp (-t) ≤ ‖x i-x j‖ ∧ ‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t)} =
{x | 1 ≤ ‖x i-x j‖} ∪
({x | Real.exp (-t) ≤ ‖x i-x j‖} ∩ {x | ‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t)}) := by
ext x
simp only [mem_ofPred_eq, mem_union, mem_inter_iff, ne_eq, hij, not_false_eq_true, imp_iff_not_or, not_lt, not_true_eq_false, false_or]
rw [he]
exact (measurableSet_le measurable_const hm).union
((measurableSet_le measurable_const hm).inter (measurableSet_le hm measurable_const))
noncomputable def freeFamily (t d : ℝ) : ConfigFamily :=
fun ι _ x => if FreeConfig t x then Real.exp (d*Fintype.card ι) else 0
lemma freeFamily_relabel (t d : ℝ) : RelabelInvariant (freeFamily t d) := by
intro ι κ _ _ e x
have he : FreeConfig t (x ∘ e) ↔ FreeConfig t x := by
constructor
· intro hh i j hij hdist
have hn : e.symm i ≠ e.symm j := fun he => hij (e.symm.injective he)
simpa only [Function.comp_apply, Equiv.apply_symm_apply] using
hh (e.symm i) (e.symm j) hn (by simpa only [Function.comp_apply, Equiv.apply_symm_apply] using hdist)
· intro hh i j hij hdist
exact hh (e i) (e j) (fun he => hij (e.injective he)) hdist
simp only [freeFamily, he, Fintype.card_congr e]
lemma freeFamily_measurable (t d : ℝ) (ι : Type) [Fintype ι] : Measurable (freeFamily t d ι) :=
Measurable.ite (measurableSet_freeConfig t) measurable_const measurable_const
lemma freeFamily_nonneg (t d : ℝ) (ι : Type) [Fintype ι] (x : ι → Space) :
0 ≤ freeFamily t d ι x := by
unfold freeFamily
split_ifs <;> positivity
lemma freeFamily_le_exp (t d : ℝ) (ι : Type) [Fintype ι] (x : ι → Space) :
freeFamily t d ι x ≤ Real.exp (d*Fintype.card ι) := by
unfold freeFamily
split_ifs
· exact le_rfl
· exact (Real.exp_pos _).le
lemma freeFamily_summable (t d L : ℝ) : Summable (familyTerm (freeFamily t d) (cube L)) :=
summable_familyTerm_of_bound _ (isCompact_cube L).measure_lt_top.ne (freeFamily_measurable t d)
(freeFamily_nonneg t d) (freeFamily_le_exp t d)
lemma rootCover_volume_real {ι : Type} [Fintype ι] (s : Finset ι) (r L : ℝ) :
volume.real ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ rootCover s r) ≤
(volume.real (cube L))^s.card *
((s.card : ℝ)*volume.real (Metric.closedBall (0 : Space) r))^(Fintype.card ι-s.card) := by
have hh := ENNReal.toReal_mono (show
(volume (cube L))^s.card *
((s.card : ℝ≥0∞)*volume (Metric.closedBall (0 : Space) r))^(Fintype.card ι-s.card) ≠ ⊤ by
exact ENNReal.mul_ne_top (ENNReal.pow_ne_top (isCompact_cube L).measure_lt_top.ne)
(ENNReal.pow_ne_top (ENNReal.mul_ne_top (by simp) (measure_closedBall_lt_top.ne)))) (rootCover_volume s r L)
simpa only [Measure.real, ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_natCast] using hh
lemma freeConfig_volume_le {ι : Type} [Fintype ι] {t : ℝ}
(ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (L : ℝ) :
volume.real ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ {x | FreeConfig t x}) ≤
∑ k ∈ Finset.range (Fintype.card ι+1),
if k ≤ packingNumber L ∧ Fintype.card ι ≤ 4*k then
((Fintype.card ι).choose k : ℝ)*(volume.real (cube L))^k *
((k : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-k)
else 0 := by
let T : Finset (Finset ι) := Finset.univ.filter (fun s => s.card ≤ packingNumber L ∧ Fintype.card ι ≤ 4*s.card)
let A : Finset ι → Set (ι → Space) := fun s =>
{x | ∀ i, x i ∈ cube L} ∩ rootCover s ((1+shellDelta)*Real.exp (-t))
have hsub : ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ {x | FreeConfig t x}) ⊆ ⋃ s ∈ T, A s := by
intro x hx
obtain ⟨s,hs,hn,hc⟩ := freeConfig_roots ht hx.2 hx.1
apply mem_iUnion₂.mpr
refine ⟨s,by simp only [T, Finset.mem_filter, Finset.mem_univ, true_and]; exact ⟨hs,hn⟩,hx.1,?_⟩
intro j
obtain ⟨i,hi,hij⟩ := hc j
exact ⟨⟨i,hi⟩,hij⟩
have hfin : volume (⋃ s ∈ T, A s) ≠ ⊤ := by
apply ne_top_of_le_ne_top (show volume {x : ι → Space | ∀ i, x i ∈ cube L} ≠ ⊤ by
rw [volume_generalConfigurations]; exact ENNReal.pow_ne_top (isCompact_cube L).measure_lt_top.ne)
apply measure_mono
intro x hx
obtain ⟨s,_,hs⟩ := mem_iUnion₂.mp hx
exact hs.1
calc
_ ≤ volume.real (⋃ s ∈ T, A s) := measureReal_mono hsub hfin
_ ≤ ∑ s ∈ T, volume.real (A s) := measureReal_biUnion_finset_le _ _
_ ≤ ∑ s ∈ T, (volume.real (cube L))^s.card *
((s.card : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-s.card) := by
exact Finset.sum_le_sum (fun s _ => rootCover_volume_real s _ L)
_ = _ := by
rw [Finset.sum_filter]
have hh := Finset.sum_powerset_apply_card
(fun k => if k ≤ packingNumber L ∧ Fintype.card ι ≤ 4*k then
(volume.real (cube L))^k *
((k : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-k) else 0)
(x := (Finset.univ : Finset ι))
simp only [Finset.powerset_univ, Finset.card_univ, nsmul_eq_mul] at hh
rw [hh]
apply Finset.sum_congr rfl
intro k hk
split_ifs <;> simp [mul_assoc]
end ContinuumTemperature
end
end OAI
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