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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 | import OAI.Probability.FirstPassage.LocalTravel
namespace OAI
namespace PlanarFPP
open Set
@[simp] theorem latticeEmbed_add (p q : Vertex) :
latticeEmbed (p+q) = latticeEmbed p+latticeEmbed q := by
ext i
fin_cases i <;> simp [latticeEmbed]
theorem step_embed_norm_le (d : Step) : ‖latticeEmbed d.delta‖ ≤ 1 := by
have h := plane_norm_le_coordinates (latticeEmbed d.delta)
cases d <;> simpa [Step.delta, latticeEmbed] using h
theorem displacement_norm_le_length (ds : List Step) :
‖latticeEmbed (displacement ds)‖ ≤ (ds.length : ℝ) := by
induction ds with
| nil => simp
| cons d ds ih =>
rw [displacement_cons, latticeEmbed_add]
have ht := norm_add_le (latticeEmbed d.delta) (latticeEmbed (displacement ds))
simp only [List.length_cons, Nat.cast_add, Nat.cast_one]
linarith [step_embed_norm_le d]
theorem coordinatePath_length (p q : Vertex) :
(coordinatePath p q).length = latticeDistance p q := by
have hh (a : ℤ) : (horizontal a).length = a.natAbs := by
unfold horizontal
split_ifs <;> simp only [List.length_replicate] <;> omega
have hv (a : ℤ) : (vertical a).length = a.natAbs := by
unfold vertical
split_ifs <;> simp only [List.length_replicate] <;> omega
simp [coordinatePath, latticeDistance, hh, hv]
theorem coordinatePath_prefix_bound (p q : Vertex) (i : ℕ) :
‖latticeEmbed (p+displacement ((coordinatePath p q).take i)-p)‖ ≤
(latticeDistance p q : ℝ) := by
simp only [add_sub_cancel_left]
exact (displacement_norm_le_length _).trans (by
exact_mod_cast (show ((coordinatePath p q).take i).length ≤ latticeDistance p q by
rw [List.length_take, coordinatePath_length]; exact min_le_right _ _))
theorem segmentPoint_perturbation (p q p' q' : Vertex) (r t : ℝ)
(ht0 : 0 ≤ t) (ht1 : t ≤ 1)
(hp : ‖latticeEmbed (p'-p)‖ ≤ r) (hq : ‖latticeEmbed (q'-q)‖ ≤ r) :
‖segmentPoint p' q' t-segmentPoint p q t‖ ≤ r := by
have he : segmentPoint p' q' t-segmentPoint p q t =
(1-t) • latticeEmbed (p'-p)+t • latticeEmbed (q'-q) := by
simp only [segmentPoint,latticeEmbed_sub]
ext i
simp [PiLp.add_apply,PiLp.sub_apply,PiLp.smul_apply,smul_eq_mul]
ring
rw [he]
have hn := norm_add_le ((1-t) • latticeEmbed (p'-p)) (t • latticeEmbed (q'-q))
rw [norm_smul,norm_smul,Real.norm_eq_abs,Real.norm_eq_abs,
abs_of_nonneg (by linarith : 0 ≤ 1-t),abs_of_nonneg ht0] at hn
have h1 := mul_le_mul_of_nonneg_left hp (show 0 ≤ 1-t by linarith)
have h2 := mul_le_mul_of_nonneg_left hq ht0
nlinarith
theorem segmentTube_perturbation (p q p' q' : Vertex) (r R : ℝ)
(hp : ‖latticeEmbed (p'-p)‖ ≤ r) (hq : ‖latticeEmbed (q'-q)‖ ≤ r) :
SegmentTube p' q' R ⊆ SegmentTube p q (R+r) := by
rintro z ⟨t,ht0,ht1,ht⟩
refine ⟨t,ht0,ht1,?_⟩
have hb := segmentPoint_perturbation p q p' q' r t ht0 ht1 hp hq
have hn := norm_add_le (latticeEmbed z-segmentPoint p' q' t)
(segmentPoint p' q' t-segmentPoint p q t)
rw [sub_add_sub_cancel] at hn
linarith
theorem localTravel_attach (μ : Plane → ℝ) (w : Edge → ℝ)
(β : ℝ) (hβ : 0 ≤ β) (hlip : ∀ v u, |μ v-μ u| ≤ β*‖v-u‖)
(p q p' q' : Vertex) (r a η δ s : ℝ)
(hp : ‖latticeEmbed (p'-p)‖ ≤ r) (hq : ‖latticeEmbed (q'-q)‖ ≤ r)
(hleft : pathCost w p (coordinatePath p p') ≤ a)
(hright : pathCost w q' (coordinatePath q' q) ≤ a)
(hmid : LocalTravel μ η δ s p' q' w) :
∃ ds, p+displacement ds=q ∧
pathCost w p ds ≤ μ (latticeEmbed (q-p))+η*s+2*a+2*β*r ∧
PathIn p ds (SegmentTube p q (max (3*r) (δ*s+r))) := by
obtain ⟨ds,hd,hcost,hin⟩ := hmid
let l := coordinatePath p p'
let v := coordinatePath q' q
have hl : p+displacement l=p' := coordinatePath_endpoint p p'
have hv : q'+displacement v=q := coordinatePath_endpoint q' q
have hμdiff : μ (latticeEmbed (q'-p')) ≤ μ (latticeEmbed (q-p))+2*β*r := by
have he : latticeEmbed (q'-p')-latticeEmbed (q-p) =
latticeEmbed (q'-q)-latticeEmbed (p'-p) := by
simp only [latticeEmbed_sub]; abel
have hn : ‖latticeEmbed (q'-p')-latticeEmbed (q-p)‖ ≤ 2*r := by
rw [he]
exact (norm_sub_le _ _).trans (by linarith)
have ht := (le_abs_self _).trans (hlip (latticeEmbed (q'-p')) (latticeEmbed (q-p)))
have hb := mul_le_mul_of_nonneg_left hn hβ
linarith
have hll : PathIn p l (SegmentTube p q (max (3*r) (δ*s+r))) := by
intro i _
refine ⟨0,by norm_num,by norm_num,?_⟩
have hb := coordinatePath_prefix_bound p p' i
have hdist := latticeDistance_le_norm p p'
rw [← latticeEmbed_sub] at hdist
have hr : 0 ≤ r := (norm_nonneg _).trans hp
simp only [segmentPoint,zero_smul,add_zero]
rw [← latticeEmbed_sub]
exact hb.trans ((hdist.trans (by linarith)).trans (le_max_left _ _))
have hrr : PathIn q' v (SegmentTube p q (max (3*r) (δ*s+r))) := by
intro i _
refine ⟨1,by norm_num,by norm_num,?_⟩
have hb := coordinatePath_prefix_bound q' q i
have hdist := latticeDistance_le_norm q' q
rw [norm_sub_rev,← latticeEmbed_sub] at hdist
have hn := norm_add_le
(latticeEmbed (q'+displacement (v.take i))-latticeEmbed q')
(latticeEmbed q'-latticeEmbed q)
rw [sub_add_sub_cancel,← latticeEmbed_sub,← latticeEmbed_sub,← latticeEmbed_sub] at hn
simp only [segmentPoint,one_smul,add_sub_cancel]
rw [← latticeEmbed_sub]
exact (hn.trans (by dsimp [v] at *; linarith)).trans (le_max_left _ _)
have hmm : PathIn p' ds (SegmentTube p q (max (3*r) (δ*s+r))) := by
intro i hi
obtain ⟨t,ht0,ht1,ht⟩ := segmentTube_perturbation p q p' q' r (δ*s) hp hq (hin i hi)
exact ⟨t,ht0,ht1,ht.trans (le_max_right _ _)⟩
refine ⟨l++ds++v,?_,?_,?_⟩
· rw [displacement_append,displacement_append,← add_assoc,← add_assoc,hl,hd,hv]
· rw [pathCost_append,pathCost_append,displacement_append,← add_assoc,hl,hd]
dsimp [l,v]
linarith
· apply pathIn_append _ _ _ _ (pathIn_append _ _ _ _ hll (by simpa [hl] using hmm))
simpa only [displacement_append,← add_assoc,hl,hd] using hrr
end PlanarFPP
end OAI
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