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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 | import OAI.Probability.FirstPassage.CrossingTrialGeometry
import OAI.Probability.FirstPassage.TiltedRegions
import OAI.Probability.FirstPassage.ConnectorEvents
import OAI.Probability.FirstPassage.SupportContactEstimate
namespace OAI
namespace PlanarFPP
open Set
def connectorWindow (b : ℝ) (X : ℤ) (Y S : ℝ) : Set Vertex :=
{p | p.1=X ∧ |transverse b p-Y| ≤ S/4}
def connectorRegion (b : ℝ) (X d : ℤ) (Y S s : ℝ) : Set Vertex :=
{p | (X : ℝ)-s ≤ p.1 ∧ (p.1 : ℝ) ≤ (X : ℝ)+d+s ∧
|transverse b p-Y| ≤ 4*S}
theorem connectorRegion_finite (b : ℝ) (X d : ℤ) (Y S s : ℝ) :
(connectorRegion b X d Y S s).Finite := by
apply (finite_tilted_rectangle b X Y (|(d : ℝ)|+|s|) (4*S)).subset
intro p hp
refine ⟨abs_le.mpr ⟨?_,?_⟩,hp.2.2⟩ <;>
linarith [hp.1,hp.2.1,le_abs_self s,le_abs_self (d : ℝ),abs_nonneg (d : ℝ)]
noncomputable def connectorVertices (b : ℝ) (X d : ℤ) (Y S s : ℝ) : Finset Vertex :=
(connectorRegion_finite b X d Y S s).toFinset
@[simp] theorem mem_connectorVertices (b : ℝ) (X d : ℤ) (Y S s : ℝ) (p : Vertex) :
p ∈ connectorVertices b X d Y S s ↔ p ∈ connectorRegion b X d Y S s :=
(connectorRegion_finite b X d Y S s).mem_toFinset
theorem connector_transverse_displacement (b : ℝ) (X d : ℤ) (Y S σ a : ℝ)
(hS : 0 ≤ S) (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2)
(p q : Vertex) (hp : p ∈ connectorWindow b X Y S)
(hq : q ∈ connectorWindow b (X+d) (Y+σ*a*S) S) :
S/2 ≤ σ*(transverse b q-transverse b p) ∧
σ*(transverse b q-transverse b p) ≤ 5*S/2 := by
have hsq : σ*σ=1 := by nlinarith [sq_abs σ]
have hep := hp.2
have heq := hq.2
have hp' : |σ*(transverse b p-Y)| ≤ S/4 := by simpa [abs_mul,hσ] using hep
have hq' : |σ*(transverse b q-(Y+σ*a*S))| ≤ S/4 := by simpa [abs_mul,hσ] using heq
have he : σ*(transverse b q-transverse b p) =
σ*(transverse b q-(Y+σ*a*S))-σ*(transverse b p-Y)+a*S := by
calc
_ = σ*(transverse b q-(Y+σ*a*S))-σ*(transverse b p-Y)+(σ*σ)*(a*S) := by ring
_ = _ := by rw [hsq]; ring
rw [he]
obtain ⟨hp₀,hp₁⟩ := abs_le.mp hp'
obtain ⟨hq₀,hq₁⟩ := abs_le.mp hq'
constructor <;> nlinarith [mul_le_mul_of_nonneg_right ha.1 hS,
mul_le_mul_of_nonneg_right ha.2 hS]
theorem connector_window_bounds (b : ℝ) (X d : ℤ) (Y S σ a : ℝ)
(hd : 0 ≤ d) (hS : 0 ≤ S) (hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2)
(p : Vertex) (hp : p ∈ connectorWindow b X Y S ∪
connectorWindow b (X+d) (Y+σ*a*S) S) :
(X : ℝ) ≤ p.1 ∧ (p.1 : ℝ) ≤ (X : ℝ)+d ∧
|transverse b p-Y| ≤ 9*S/4 := by
have hd' : (0 : ℝ) ≤ d := by exact_mod_cast hd
rcases hp with hp | hp
· refine ⟨by simp [hp.1],?_,?_⟩
· simp only [hp.1]; linarith
· linarith [hp.2]
· refine ⟨?_,by simp [hp.1],?_⟩
· simp only [hp.1,Int.cast_add]; linarith
· have hh := abs_add_le (transverse b p-(Y+σ*a*S)) (σ*a*S)
have he : transverse b p-(Y+σ*a*S)+σ*a*S=transverse b p-Y := by ring
rw [he,abs_mul,abs_mul,hσ,one_mul,abs_of_nonneg (by linarith : 0 ≤ a),abs_of_nonneg hS] at hh
nlinarith [hp.2,mul_le_mul_of_nonneg_right ha.2 hS]
theorem transverse_plane_lipschitz (b : ℝ) (hb : |b| ≤ 1) (z : Plane) :
|rCoord b z| ≤ 2*‖z‖ := by
have hh := abs_sub_le (z 1) 0 (b*z 0)
simp only [sub_zero,zero_sub,abs_neg,abs_mul] at hh
have hc₀ := plane_coordinate_le_norm z 0
have hc₁ := plane_coordinate_le_norm z 1
have hm := mul_le_mul_of_nonneg_right hb (abs_nonneg (z 0))
change |z 1-b*z 0| ≤ _
nlinarith
theorem plane_norm_le_shear (b : ℝ) (hb : |b| ≤ 1) (z : Plane) :
‖z‖ ≤ 2*|xCoord z|+|rCoord b z| := by
have hh := abs_add_le (rCoord b z) (b*z 0)
have he : rCoord b z+b*z 0=z 1 := by simp [rCoord_apply]
rw [he,abs_mul] at hh
have hm := mul_le_mul_of_nonneg_right hb (abs_nonneg (z 0))
have hn := plane_norm_le_coordinates z
change ‖z‖ ≤ 2*|z 0|+|rCoord b z|
linarith
theorem connector_union_diameter (b : ℝ) (hb : |b| ≤ 1)
(X d : ℤ) (Y S σ a : ℝ) (hd : 0 ≤ d) (hS : 0 ≤ S)
(hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2)
(p q : Vertex)
(hp : p ∈ connectorWindow b X Y S ∪ connectorWindow b (X+d) (Y+σ*a*S) S)
(hq : q ∈ connectorWindow b X Y S ∪ connectorWindow b (X+d) (Y+σ*a*S) S) :
‖latticeEmbed (q-p)‖ ≤ 2*(d : ℝ)+9*S/2 := by
obtain ⟨hp₀,hp₁,hp₂⟩ := connector_window_bounds b X d Y S σ a hd hS hσ ha p hp
obtain ⟨hq₀,hq₁,hq₂⟩ := connector_window_bounds b X d Y S σ a hd hS hσ ha q hq
have hx : |xCoord (latticeEmbed (q-p))| ≤ (d : ℝ) := by
change |((q-p).1 : ℝ)| ≤ _
simp only [Prod.fst_sub,Int.cast_sub]
exact abs_le.mpr ⟨by linarith,by linarith⟩
have hr : |rCoord b (latticeEmbed (q-p))| ≤ 9*S/2 := by
have hh := abs_sub_le (transverse b q-Y) 0 (transverse b p-Y)
simp only [sub_zero,zero_sub,abs_neg] at hh
have he : rCoord b (latticeEmbed (q-p))=(transverse b q-Y)-(transverse b p-Y) := by
simp [rCoord_apply,latticeEmbed,transverse]; ring
rw [he]
linarith
exact (plane_norm_le_shear b hb _).trans (by linarith)
theorem connector_tube_containment (b : ℝ) (hb : |b| ≤ 1)
(X d : ℤ) (Y S s σ a : ℝ) (hd : 0 ≤ d) (hs : 0 ≤ s) (hS : s ≤ S)
(hσ : |σ|=1) (ha : 1 ≤ a ∧ a ≤ 2)
(p q : Vertex) (hp : p ∈ connectorWindow b X Y S)
(hq : q ∈ connectorWindow b (X+d) (Y+σ*a*S) S) :
SegmentTube p q (s/2) ⊆ (connectorVertices b X d Y S s : Set Vertex) := by
have hS0 : 0 ≤ S := hs.trans hS
obtain ⟨hp₀,hp₁,hp₂⟩ := connector_window_bounds b X d Y S σ a hd hS0 hσ ha p (Or.inl hp)
obtain ⟨hq₀,hq₁,hq₂⟩ := connector_window_bounds b X d Y S σ a hd hS0 hσ ha q (Or.inr hq)
intro z hz
obtain ⟨t,ht₀,ht₁,ht⟩ := hz
have hsegx : xCoord (segmentPoint p q t)=(1-t)*(p.1 : ℝ)+t*(q.1 : ℝ) := by
simp [segmentPoint,latticeEmbed,xCoord_apply,PiLp.add_apply,PiLp.sub_apply,PiLp.smul_apply]
ring
have hsegr : rCoord b (segmentPoint p q t)-Y=
(1-t)*(transverse b p-Y)+t*(transverse b q-Y) := by
simp [segmentPoint,latticeEmbed,rCoord_apply,transverse,PiLp.add_apply,PiLp.sub_apply,PiLp.smul_apply]
ring
have hseg₀ : (X : ℝ) ≤ xCoord (segmentPoint p q t) := by
rw [hsegx]
nlinarith [mul_le_mul_of_nonneg_left hp₀ (by linarith : 0 ≤ 1-t),
mul_le_mul_of_nonneg_left hq₀ ht₀]
have hseg₁ : xCoord (segmentPoint p q t) ≤ (X : ℝ)+d := by
rw [hsegx]
nlinarith [mul_le_mul_of_nonneg_left hp₁ (by linarith : 0 ≤ 1-t),
mul_le_mul_of_nonneg_left hq₁ ht₀]
have hseg₂ : |rCoord b (segmentPoint p q t)-Y| ≤ 9*S/4 := by
rw [hsegr]
have hh := abs_add_le ((1-t)*(transverse b p-Y)) (t*(transverse b q-Y))
rw [abs_mul,abs_mul,abs_of_nonneg ht₀,abs_of_nonneg (by linarith : 0 ≤ 1-t)] at hh
nlinarith [mul_le_mul_of_nonneg_left hp₂ (by linarith : 0 ≤ 1-t),
mul_le_mul_of_nonneg_left hq₂ ht₀]
have hx := (plane_coordinate_le_norm (latticeEmbed z-segmentPoint p q t) 0).trans ht
have hr := (transverse_plane_lipschitz b hb (latticeEmbed z-segmentPoint p q t)).trans
(mul_le_mul_of_nonneg_left ht (by norm_num : (0 : ℝ) ≤ 2))
have hx' : |(z.1 : ℝ)-xCoord (segmentPoint p q t)| ≤ s/2 := by
simpa [latticeEmbed,PiLp.sub_apply] using hx
have hr' : |transverse b z-rCoord b (segmentPoint p q t)| ≤ s := by
have he : 2*(s/2)=s := by ring
simpa [map_sub,rCoord_apply,latticeEmbed,transverse,he] using hr
change z ∈ connectorVertices b X d Y S s
rw [mem_connectorVertices]
refine ⟨?_,?_,?_⟩
· linarith [(abs_le.mp hx').1]
· linarith [(abs_le.mp hx').2]
· have hh := abs_add_le (transverse b z-rCoord b (segmentPoint p q t))
(rCoord b (segmentPoint p q t)-Y)
rw [sub_add_sub_cancel] at hh
linarith
end PlanarFPP
end OAI
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