| 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 |
|
|
