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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 | import OAI.Probability.FirstPassage.EndpointNet
namespace OAI
namespace PlanarFPP
open Set Filter Finset MeasureTheory ProbabilityTheory
open scoped Topology
def BallTravel (μ : Plane → ℝ) (D η δ s : ℝ) (w : Edge → ℝ) : Prop :=
∀ p q : Vertex, ‖latticeEmbed p‖ ≤ D*s → ‖latticeEmbed q‖ ≤ D*s →
LocalTravel μ η δ s p q w
theorem ballTravel_of_net (μ : Plane → ℝ) (w : Edge → ℝ)
(D γ β η δ κ : ℝ) (hβ : 0 ≤ β)
(hlip : ∀ v u, |μ v-μ u| ≤ β*‖v-u‖)
(S : Finset {v : Plane // ‖v‖ ≤ D})
(hS : ∀ v : Plane, ‖v‖ ≤ D → ∃ u ∈ S, ‖v-u.val‖ < γ)
(C n : ℕ) (hn : 0 < n) (hbox : D*n+2 ≤ (C*n : ℕ))
(hcost : η/4*n+2*(8*(γ*n+2)+κ*(C*n : ℕ))+2*β*(γ*n+2) ≤ η*n)
(htube : max (3*(γ*n+2)) (δ/2*n+(γ*n+2)) ≤ δ*n)
(hc : ∀ p ∈ latticeBox (C*n), ∀ q ∈ latticeBox (C*n),
pathCost w p (coordinatePath p q) < 4*(latticeDistance p q : ℝ)+κ*(C*n : ℕ))
(hm : ∀ u ∈ S, ∀ v ∈ S, LocalTravel μ (η/4) (δ/2) n
(latticeFloor ((n : ℝ) • u.val)) (latticeFloor ((n : ℝ) • v.val)) w) :
BallTravel μ D η δ n w := by
intro p q hp hq
obtain ⟨u,hu,hpu⟩ := scaled_net_approximates D γ S hS n hn p hp
obtain ⟨v,hv,hqv⟩ := scaled_net_approximates D γ S hS n hn q hq
let p' := latticeFloor ((n : ℝ) • u.val)
let q' := latticeFloor ((n : ℝ) • v.val)
have hpbox : p ∈ latticeBox (C*n) := lattice_mem_box_of_norm p (C*n) (hp.trans (by linarith))
have hqbox : q ∈ latticeBox (C*n) := lattice_mem_box_of_norm q (C*n) (hq.trans (by linarith))
have hp'box : p' ∈ latticeBox (C*n) := lattice_mem_box_of_norm p' (C*n)
((scaled_net_norm D u n).trans hbox)
have hq'box : q' ∈ latticeBox (C*n) := lattice_mem_box_of_norm q' (C*n)
((scaled_net_norm D v n).trans hbox)
have hleft : pathCost w p (coordinatePath p p') ≤ 8*(γ*n+2)+κ*(C*n : ℕ) := by
have hd := latticeDistance_le_norm p p'
rw [← latticeEmbed_sub] at hd
have hh := hc p hpbox p' hp'box
dsimp [p'] at *
linarith
have hright : pathCost w q' (coordinatePath q' q) ≤ 8*(γ*n+2)+κ*(C*n : ℕ) := by
have hd := latticeDistance_le_norm q' q
rw [norm_sub_rev,← latticeEmbed_sub] at hd
have hh := hc q' hq'box q hqbox
dsimp [q'] at *
linarith
obtain ⟨ds,hend,hc',ht'⟩ := localTravel_attach μ w β hβ hlip p q p' q'
(γ*n+2) (8*(γ*n+2)+κ*(C*n : ℕ)) (η/4) (δ/2) n hpu hqv hleft hright (hm u hu v hv)
refine ⟨ds,hend,hc'.trans (by linarith),?_⟩
intro i hi
obtain ⟨t,ht0,ht1,hbound⟩ := ht' i hi
exact ⟨t,ht0,ht1,hbound.trans htube⟩
theorem ballTravel_parameters (D β η δ : ℝ)
(hD : 0 < D) (hβ : 0 ≤ β) (hη : 0 < η) (hδ : 0 < δ) :
∃ γ κ : ℝ, ∃ C : ℕ, 0 < γ ∧ 0 < κ ∧ 0 < C ∧
∀ᶠ n : ℕ in atTop,
0 < n ∧ D*n+2 ≤ (C*n : ℕ) ∧
η/4*n+2*(8*(γ*n+2)+κ*(C*n : ℕ))+2*β*(γ*n+2) ≤ η*n ∧
max (3*(γ*n+2)) (δ/2*n+(γ*n+2)) ≤ δ*n := by
obtain ⟨C,hC⟩ := exists_nat_gt (D+2)
have hc : (0 : ℝ) < C := by linarith
have hc' : 0 < C := by exact_mod_cast hc
let γ := min (η/(8*(16+2*β))) (δ/12)
let κ := η/(16*(C : ℝ))
have hγ : 0 < γ := lt_min (div_pos hη (by positivity)) (by positivity)
have hκ : 0 < κ := div_pos hη (by positivity)
have hgη : (16+2*β)*γ ≤ η/8 := by
have ht := (le_div_iff₀ (by positivity : 0 < 8*(16+2*β))).mp (min_le_left _ _ : γ ≤ _)
nlinarith
have hgδ : γ ≤ δ/12 := min_le_right _ _
have hk : κ*(C : ℝ)=η/16 := by dsimp [κ]; field_simp
refine ⟨γ,κ,C,hγ,hκ,hc',?_⟩
have hlim := (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop)
filter_upwards [eventually_ge_atTop 1,
hlim.eventually (eventually_ge_atTop (4*(32+4*β)/η)),
hlim.eventually (eventually_ge_atTop (24/δ))] with n hn hnη hnδ
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hn0 : (0 : ℝ) ≤ n := by positivity
have hhη := (div_le_iff₀ hη).mp hnη
have hhδ := (div_le_iff₀ hδ).mp hnδ
have hge := mul_le_mul_of_nonneg_right hgη hn0
have hgd := mul_le_mul_of_nonneg_right hgδ hn0
have hcb := mul_le_mul_of_nonneg_right hC.le hn0
simp only [Nat.cast_mul]
refine ⟨by omega,by nlinarith,?_,?_⟩
· have he : κ*((C : ℝ)*n)=(η/16)*n := by rw [← mul_assoc,hk]
rw [he]
nlinarith
· apply max_le <;> nlinarith
variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ}
theorem badConnections_tendsto_zero (h : ExponentialEnvironment P τ)
(κ : ℝ) (hκ : 0 < κ) :
Tendsto (fun n => P.real (badConnections τ κ n)) atTop (𝓝 0) := by
apply squeeze_zero (fun _ => measureReal_nonneg) (badConnections_probability h κ)
exact (summable_box_connection_bound hκ).tendsto_atTop_zero
theorem exponential_ball_travel_nat (h : ExponentialEnvironment P τ)
(μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
(D η δ : ℝ) (hD : 0 < D) (hη : 0 < η) (hδ : 0 < δ) :
Tendsto (fun n : ℕ => P.real {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)})
atTop (𝓝 0) := by
classical
let _ := h.probability
obtain ⟨β,hβ,hlip⟩ := timeConstantNorm_lipschitz_bound μ hμ
obtain ⟨γ,κ,C,hγ,hκ,hC,hparams⟩ := ballTravel_parameters D β η δ hD hβ.le hη hδ
obtain ⟨S,hS⟩ := finite_ball_net D γ hγ
let A (n : ℕ) := ⋃ uv ∈ S ×ˢ S, {ω | ¬LocalTravel μ (η/4) (δ/2) n
(latticeFloor ((n : ℝ) • uv.1.val)) (latticeFloor ((n : ℝ) • uv.2.val)) (fun e => τ e ω)}
have hA : Tendsto (fun n => P.real (A n)) atTop (𝓝 0) := by
apply (tendsto_order.2 ⟨?_,?_⟩)
· intro a ha
exact Eventually.of_forall (fun _ => ha.trans_le measureReal_nonneg)
· intro ε hε
have hden : 0 < ((S ×ˢ S).card : ℝ)+1 := by positivity
filter_upwards [eventually_ge_atTop 1,
exponential_local_travel_nat h μ hμ ht (2*D+4) (η/4) (δ/2)
(by linarith) (by linarith) (by linarith)
(ε/(((S ×ˢ S).card : ℝ)+1)) (div_pos hε hden)] with n hn hh
have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn
have hbound : P.real (A n) ≤ ((S ×ˢ S).card : ℝ)*(ε/(((S ×ˢ S).card : ℝ)+1)) := by
refine (measureReal_biUnion_finset_le _ _).trans ?_
calc
_ ≤ ∑ _uv ∈ S ×ˢ S, ε/(((S ×ˢ S).card : ℝ)+1) := by
apply sum_le_sum
intro uv huv
apply (hh _ _ ?_).le
rw [latticeEmbed_sub]
exact (norm_sub_le _ _).trans (by
have h1 := scaled_net_norm D uv.1 n
have h2 := scaled_net_norm D uv.2 n
nlinarith)
_ = _ := by simp
apply hbound.trans_lt
have he := (div_pos hε hden)
have hid : ((((S ×ˢ S).card : ℝ)+1))*(ε/(((S ×ˢ S).card : ℝ)+1))=ε := by field_simp
nlinarith
have hCn : Tendsto (fun n : ℕ => C*n) atTop atTop := by
apply tendsto_atTop.mpr
intro b
filter_upwards [eventually_ge_atTop b] with n hn
exact hn.trans (Nat.le_mul_of_pos_left n hC)
have hB := (badConnections_tendsto_zero h κ hκ).comp hCn
have hbound : ∀ᶠ n : ℕ in atTop,
P.real {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)} ≤
P.real (badConnections τ κ (C*n))+P.real (A n) := by
filter_upwards [hparams] with n hn
have hsub : {ω | ¬BallTravel μ D η δ n (fun e => τ e ω)} ⊆
badConnections τ κ (C*n) ∪ A n := by
intro ω hω
by_contra hb
have hc : ω ∉ badConnections τ κ (C*n) := fun hc => hb (Or.inl hc)
have hm : ω ∉ A n := fun hm => hb (Or.inr hm)
apply hω
apply ballTravel_of_net μ _ D γ β η δ κ hβ.le hlip S hS C n
hn.1 hn.2.1 hn.2.2.1 hn.2.2.2
· intro p hp q hq
by_contra hh
exact hc (mem_iUnion₂.mpr ⟨(p,q),mem_product.mpr ⟨hp,hq⟩,le_of_not_gt hh⟩)
· intro u hu v hv
by_contra hh
exact hm (mem_iUnion₂.mpr ⟨(u,v),mem_product.mpr ⟨hu,hv⟩,hh⟩)
exact (measureReal_mono hsub).trans (measureReal_union_le _ _)
exact squeeze_zero' (Eventually.of_forall (fun _ => measureReal_nonneg)) hbound
(by simpa using hB.add hA)
end PlanarFPP
end OAI
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