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b317718 | 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 | import OAI.Probability.GammaPassage.GridIndependence
import OAI.Probability.GammaPassage.OriginEscape
namespace OAI
namespace GammaFPP.CrossingBridge
open PlanarFPP Set Finset MeasureTheory ProbabilityTheory
abbrev PathIn := IntervalTravelBridge.PathIn
abbrev DependsOnEdges := Grid.DependsOnEdges
@[simp] theorem pathIn_iff_interval (p : Vertex) (ds : List Step) (S : Set Vertex) :
PathIn p ds S ↔ IntervalTravelBridge.PathIn p ds S := Iff.rfl
@[simp] theorem depends_iff_grid (A : Set (Edge → ℝ)) (S : Finset Edge) :
DependsOnEdges A S ↔ Grid.DependsOnEdges A S := Iff.rfl
def edgeEnd (e : Edge) : Vertex := e.1 + if e.2 then (0,1) else (1,0)
def internalEdges (V : Finset Vertex) : Finset Edge :=
(V ×ˢ Finset.univ).filter (fun e => edgeEnd e ∈ V)
theorem mem_internalEdges (e : Edge) (V : Finset Vertex) :
e ∈ internalEdges V ↔ e.1 ∈ V ∧ edgeEnd e ∈ V := by
simp [internalEdges]
theorem internalEdges_card_le (V : Finset Vertex) : (internalEdges V).card ≤ 2*V.card := by
have h := Finset.card_filter_le (s := V ×ˢ (Finset.univ : Finset Bool)) (fun e => edgeEnd e ∈ V)
simpa [internalEdges,Nat.mul_comm] using h
theorem step_mem_internalEdges (V : Finset Vertex) (p : Vertex) (d : Step)
(hp : p ∈ V) (hq : p+d.delta ∈ V) : d.edge p ∈ internalEdges V := by
rw [mem_internalEdges]
cases d <;> simp_all [Step.edge,Step.delta,edgeEnd,add_assoc]
all_goals change p+0 ∈ V; simpa using hp
theorem internalEdges_disjoint {V W : Finset Vertex} (h : Disjoint V W) :
Disjoint (internalEdges V) (internalEdges W) := by
apply Finset.disjoint_left.mpr
intro e he hf
exact Finset.disjoint_left.mp h (mem_internalEdges e V |>.mp he).1
(mem_internalEdges e W |>.mp hf).1
def pathEdges (p : Vertex) : List Step → Finset Edge
| [] => ∅
| d::ds => insert (d.edge p) (pathEdges (p+d.delta) ds)
theorem pathIn_cons_iff (p : Vertex) (d : Step) (ds : List Step) (S : Set Vertex) :
PathIn p (d::ds) S ↔ p ∈ S ∧ PathIn (p+d.delta) ds S := by
constructor
· intro h
refine ⟨by simpa using h 0 (by simp),?_⟩
intro i hi
have hh := h (i+1) (by simpa using hi)
simpa only [List.take_succ_cons,displacement_cons,add_assoc] using hh
· rintro ⟨hp,ht⟩ i hi
cases i with
| zero => simpa using hp
| succ i =>
have hh := ht i (by simpa using hi)
simpa only [List.take_succ_cons,displacement_cons,add_assoc] using hh
theorem pathIn_endpoint (p : Vertex) (ds : List Step) (S : Set Vertex)
(h : PathIn p ds S) : p+displacement ds ∈ S := by
simpa using h ds.length le_rfl
theorem pathEdges_subset_internalEdges (V : Finset Vertex) (p : Vertex) (ds : List Step)
(h : PathIn p ds (V : Set Vertex)) : pathEdges p ds ⊆ internalEdges V := by
induction ds generalizing p with
| nil => simp [pathEdges]
| cons d ds ih =>
obtain ⟨hp,ht⟩ := (pathIn_cons_iff p d ds _).mp h
have hq : p+d.delta ∈ V := by simpa using ht 0 (by omega)
intro e he
rcases Finset.mem_insert.mp he with he | he
· rw [he]
exact step_mem_internalEdges V p d hp hq
· exact ih (p+d.delta) ht he
theorem pathCost_eq_of_agree (w w' : Edge → ℝ) (p : Vertex) (ds : List Step)
(h : ∀ e ∈ pathEdges p ds, w e=w' e) : pathCost w p ds=pathCost w' p ds := by
induction ds generalizing p with
| nil => rfl
| cons d ds ih =>
simp only [pathCost]
rw [h (d.edge p) (by simp [pathEdges])]
rw [ih (p+d.delta) (fun e he => h e (by simp [pathEdges,he]))]
def strictLocalizedEvent (V : Finset Vertex) (family : Vertex → List Step → Prop)
(bound : Vertex → List Step → ℝ) : Set (Edge → ℝ) :=
{w | ∃ p ds, family p ds ∧ PathIn p ds (V : Set Vertex) ∧ pathCost w p ds < bound p ds}
def boundedLocalizedEvent (V : Finset Vertex) (family : Vertex → List Step → Prop)
(bound : Vertex → List Step → ℝ) : Set (Edge → ℝ) :=
{w | ∃ p ds, family p ds ∧ PathIn p ds (V : Set Vertex) ∧ pathCost w p ds ≤ bound p ds}
theorem measurableSet_strictLocalizedEvent (V : Finset Vertex)
(family : Vertex → List Step → Prop) (bound : Vertex → List Step → ℝ) :
MeasurableSet (strictLocalizedEvent V family bound) := by
simpa only [strictLocalizedEvent,and_assoc] using
measurableSet_strict_path_event (τ := fun e (w : Edge → ℝ) => w e) measurable_pi_apply
(fun p ds => family p ds ∧ PathIn p ds (V : Set Vertex)) bound
theorem measurableSet_boundedLocalizedEvent (V : Finset Vertex)
(family : Vertex → List Step → Prop) (bound : Vertex → List Step → ℝ) :
MeasurableSet (boundedLocalizedEvent V family bound) := by
simpa only [boundedLocalizedEvent,and_assoc] using
measurableSet_bounded_path_event (τ := fun e (w : Edge → ℝ) => w e) measurable_pi_apply
(fun p ds => family p ds ∧ PathIn p ds (V : Set Vertex)) bound
theorem strictLocalizedEvent_depends (V : Finset Vertex)
(family : Vertex → List Step → Prop) (bound : Vertex → List Step → ℝ) :
DependsOnEdges (strictLocalizedEvent V family bound) (internalEdges V) := by
intro w w' hh
have hc (p ds) (hin : PathIn p ds (V : Set Vertex)) : pathCost w p ds=pathCost w' p ds :=
pathCost_eq_of_agree w w' p ds (fun e he => hh e (pathEdges_subset_internalEdges V p ds hin he))
constructor
· rintro ⟨p,ds,hf,hin,hcost⟩
exact ⟨p,ds,hf,hin,by rwa [← hc p ds hin]⟩
· rintro ⟨p,ds,hf,hin,hcost⟩
exact ⟨p,ds,hf,hin,by rwa [hc p ds hin]⟩
theorem boundedLocalizedEvent_depends (V : Finset Vertex)
(family : Vertex → List Step → Prop) (bound : Vertex → List Step → ℝ) :
DependsOnEdges (boundedLocalizedEvent V family bound) (internalEdges V) := by
intro w w' hh
have hc (p ds) (hin : PathIn p ds (V : Set Vertex)) : pathCost w p ds=pathCost w' p ds :=
pathCost_eq_of_agree w w' p ds (fun e he => hh e (pathEdges_subset_internalEdges V p ds hin he))
constructor
· rintro ⟨p,ds,hf,hin,hcost⟩
exact ⟨p,ds,hf,hin,by rwa [← hc p ds hin]⟩
· rintro ⟨p,ds,hf,hin,hcost⟩
exact ⟨p,ds,hf,hin,by rwa [hc p ds hin]⟩
theorem strip_vertices_disjoint (V W : Finset Vertex) (b y z R : ℝ)
(hV : ∀ v ∈ V, |transverse b v-y| ≤ R)
(hW : ∀ v ∈ W, |transverse b v-z| ≤ R)
(hsep : 2*R < |y-z|) : Disjoint V W := by
apply Finset.disjoint_left.mpr
intro v hv hw
have ht := abs_add_le (y-transverse b v) (transverse b v-z)
rw [sub_add_sub_cancel,abs_sub_comm y (transverse b v)] at ht
have h1 := hV v hv
have h2 := hW v hw
linarith
end GammaFPP.CrossingBridge
end OAI
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