File size: 4,403 Bytes
dd14c26 | 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 | import OAI.Probability.LatticePassage.FiniteResampling
import OAI.Probability.LatticePassage.ExponentialIncrement
namespace OAI
namespace CompanionFPP
open MeasureTheory ProbabilityTheory Real Finset
open scoped ENNReal
theorem sum_upwardResampling_transform (ω : Environment) (hω : ∀ e, 0 ≤ ω e)
(x y : Vertex) (R : ℕ) (hR : latticeDistance x y < R)
(φ : ℝ → ℝ≥0∞) (hφ : MonotoneOn φ (Set.Ici 0)) (hzero : φ 0 = 0) :
(∑ e ∈ localEdges x R, ∫⁻ u, φ (upwardResampling ω x y R e u) ∂expMeasure 1) ≤
(∫⁻ u, φ u ∂expMeasure 1) * R := by
classical
obtain ⟨p,hp,hl,hm⟩ := local_geodesic_exists ω x y R hR
have hsub : (pathEdges p).toFinset ⊆ localEdges x R := by
intro e he
exact mem_biUnion.mpr ⟨⟨y,p⟩, mem_localCandidates p hp hl, he⟩
have he : ∀ e, (∫⁻ u, φ (upwardResampling ω x y R e u) ∂expMeasure 1) ≤
if e ∈ pathEdges p then ∫⁻ u, φ u ∂expMeasure 1 else 0 := by
intro e
by_cases he : e ∈ pathEdges p
· rw [ite_eq_left he]
apply lintegral_mono_ae
filter_upwards [exponential_positive] with u hu
apply hφ (by change 0 ≤ upwardResampling ω x y R e u; unfold upwardResampling; positivity) hu.le
simpa only [ite_eq_left he] using upwardResampling_le ω hω p hp hl hm e u hu.le
· rw [ite_eq_right he, ← lintegral_zero (μ := expMeasure 1)]
apply lintegral_mono_ae
filter_upwards [exponential_positive] with u hu
have hb := upwardResampling_le ω hω p hp hl hm e u hu.le
simp only [ite_eq_right he] at hb
have hn : 0 ≤ upwardResampling ω x y R e u := by unfold upwardResampling; positivity
rw [le_antisymm hb hn, hzero]
calc
_ ≤ ∑ e ∈ localEdges x R, if e ∈ pathEdges p then ∫⁻ u, φ u ∂expMeasure 1 else 0 :=
sum_le_sum (fun e _ => he e)
_ = ∑ e ∈ (pathEdges p).toFinset, ∫⁻ u, φ u ∂expMeasure 1 := by
rw [← sum_filter]
congr 1
ext e
simp only [mem_filter, List.mem_toFinset]
exact and_iff_right_of_imp (fun he => hsub (List.mem_toFinset.mpr he))
_ = (p.length : ℝ≥0∞) * ∫⁻ u, φ u ∂expMeasure 1 := by
rw [sum_const, nsmul_eq_mul, List.toFinset_card_of_nodup (pathEdges_nodup hp.isTrail),
pathEdges_length]
_ ≤ (∫⁻ u, φ u ∂expMeasure 1)*R := by
rw [mul_comm]
gcongr
lemma exponential_increment_monotone (a : ℝ) :
MonotoneOn (fun u => ENNReal.ofReal ((exp (a*u/2)-1)^2)) (Set.Ici 0) := by
intro u hu v hv huv
change 0 ≤ u at hu
change 0 ≤ v at hv
apply ENNReal.ofReal_le_ofReal
rcases le_total 0 a with ha | ha
· have he : exp (a*u/2) ≤ exp (a*v/2) := exp_le_exp.mpr (by nlinarith)
have h1 : 1 ≤ exp (a*u/2) := one_le_exp_iff.mpr (by nlinarith)
nlinarith
· have he : exp (a*v/2) ≤ exp (a*u/2) := exp_le_exp.mpr (by nlinarith)
have h1 : exp (a*u/2) ≤ 1 := exp_le_one_iff.mpr (by nlinarith)
nlinarith
theorem sum_upwardResampling_exponential (ω : Environment) (hω : ∀ e, 0 ≤ ω e)
(x y : Vertex) (R : ℕ) (hR : latticeDistance x y < R)
(a : ℝ) (ha : |a| ≤ 1/2) :
(∑ e ∈ localEdges x R, ∫⁻ u, ENNReal.ofReal
((exp (a * upwardResampling ω x y R e u / 2)-1)^2) ∂expMeasure 1) ≤
ENNReal.ofReal (2*a^2) * R := by
apply (sum_upwardResampling_transform ω hω x y R hR _ (exponential_increment_monotone a)
(by simp)).trans
gcongr
exact exponential_half_increment_square_le a ha
theorem finite_upward_exponential_sum (x y : Vertex) (R : ℕ)
(hR : latticeDistance x y < R) (v : LocalCoordinates x R) (hv : ∀ e, 0 ≤ v e)
(a : ℝ) (ha : |a| ≤ 1/2) :
(∑ e : ↥(localEdges x R), ∫⁻ u, ENNReal.ofReal
((exp (a * max (finitePassageTime x y R (Function.update v e u) -
finitePassageTime x y R v) 0 / 2)-1)^2) ∂expMeasure 1) ≤
ENNReal.ofReal (2*a^2) * R := by
classical
have hn : ∀ e, 0 ≤ extendLocalCoordinates x R v e := by
intro e
unfold extendLocalCoordinates
split_ifs with he
· exact hv ⟨e,he⟩
· exact le_rfl
have h := sum_upwardResampling_exponential (extendLocalCoordinates x R v) hn x y R hR a ha
rw [← sum_attach (localEdges x R)] at h
simpa only [attach_eq_univ, finitePassageTime, extendLocalCoordinates_update, upwardResampling]
using h
end CompanionFPP
end OAI
|