File size: 4,795 Bytes
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 | import OAI.Probability.FirstPassage.ConnectorGeometry
namespace OAI
namespace PlanarFPP
open Set Filter MeasureTheory
open scoped Topology
def contactSign (plus : Bool) : ℝ := if plus then 1 else -1
@[simp] theorem abs_contactSign (plus : Bool) : |contactSign plus|=1 := by
cases plus <;> norm_num [contactSign]
theorem contactSide_of_sign (plus : Bool) (r : ℝ) (hr : 0 < contactSign plus*r) :
r ∈ contactSide plus := by
cases plus <;> simpa [contactSign,contactSide] using hr
theorem trialWindow_scale_bounds (M : ℕ) (hM : 0 < M) (R s : ℝ)
(hR : 0 ≤ R) (hs : 1 ≤ s) :
s ≤ trialWindow M R s ∧ trialWindow M R s ≤ (200*M*(R+1))*s := by
have hM' : (1 : ℝ) ≤ M := by exact_mod_cast hM
have hc := Nat.le_ceil s
have hc' := Nat.ceil_lt_add_one (by linarith : 0 ≤ s)
have hbase : s ≤ (R+1)*(⌈s⌉₊ : ℝ) := by nlinarith
have hprod := mul_le_mul_of_nonneg_right hM' (by positivity : 0 ≤ (R+1)*(⌈s⌉₊ : ℝ))
have hupper := mul_le_mul_of_nonneg_left (show (⌈s⌉₊ : ℝ) ≤ 2*s by linarith)
(show 0 ≤ 100*(M : ℝ)*(R+1) by positivity)
unfold trialWindow trialSpacing
constructor <;> nlinarith
noncomputable def sourceSupportCost (μ : Plane → ℝ) (b : ℝ) (plus : Bool)
(ε s : ℝ) (p q : Vertex) : ℝ :=
μ (direction b)*xCoord (latticeEmbed (q-p))+
extremeSlope μ b plus*rCoord b (latticeEmbed (q-p))+ε*s/2
theorem exponential_source_connector_test {Ω : Type*} [MeasurableSpace Ω]
{P : Measure Ω} {τ : Edge → Ω → ℝ} (h : ExponentialEnvironment P τ)
(μ : Plane → ℝ) (hμ : TimeConstantNorm μ) (ht : IsTimeConstant P τ μ)
(b : ℝ) (hb : |b| ≤ 1) (plus : Bool) (M : ℕ) (hM : 0 < M)
(R ε : ℝ) (hR : 0 ≤ R) (hε : 0 < ε) :
∃ D0 : ℝ, 1 ≤ D0 ∧ ∀ κ > 0, ∀ᶠ s : ℝ in atTop,
∀ X : ℤ, ∀ Y a : ℝ, 1 ≤ a → a ≤ 2 →
P.real {ω | (fun e => τ e ω) ∉ allPairsRegionEvent
(connectorVertices b X ⌈D0*s⌉ Y (trialWindow M R s) s)
(connectorWindow b X Y (trialWindow M R s))
(connectorWindow b (X+⌈D0*s⌉)
(Y+contactSign plus*a*trialWindow M R s) (trialWindow M R s))
(sourceSupportCost μ b plus ε s)} < κ := by
let C : ℝ := 200*M*(R+1)
have hM' : (0 : ℝ) < M := by exact_mod_cast hM
have hC : 0 < C := by dsimp [C]; positivity
obtain ⟨D0,hD0,hgap⟩ := connector_support_gap μ hμ b plus ε C hε hC
let D := 2*(D0+1)+9*C/2
have hD : 0 < D := by dsimp [D]; linarith
refine ⟨D0,hD0,?_⟩
intro κ hκ
filter_upwards [exponential_allPairsRegionEvent h μ hμ ht D (ε/4) (1/2)
hD (by positivity) (by norm_num) κ hκ,eventually_ge_atTop (1 : ℝ)] with s hs hs1
intro X Y a ha₀ ha₁
let S := trialWindow M R s
let d : ℤ := ⌈D0*s⌉
have hs0 : 0 < s := by linarith
have hSb := trialWindow_scale_bounds M hM R s hR hs1
have hS : s ≤ S := hSb.1
have hSC : S ≤ C*s := hSb.2
have hS0 : 0 < S := hs0.trans_le hS
have hdlo : D0*s ≤ (d : ℝ) := Int.le_ceil _
have hdhi : (d : ℝ) ≤ (D0+1)*s := by
have hh : (d : ℝ) < D0*s+1 := Int.ceil_lt_add_one _
nlinarith
have hd0 : 0 ≤ d := by
have : (0 : ℝ) ≤ d := by nlinarith
exact_mod_cast this
apply hs
· intro p hp q hq
exact (connector_union_diameter b hb X d Y S (contactSign plus) a hd0 hS0.le
(abs_contactSign plus) ⟨ha₀,ha₁⟩ p q hp hq).trans (by dsimp [D]; nlinarith)
· intro p hp q hq
have hx : xCoord (latticeEmbed (q-p))=(d : ℝ) := by
change ((q-p).1 : ℝ)=_
rw [Prod.fst_sub,hp.1,hq.1]
congr 1
omega
have hr : rCoord b (latticeEmbed (q-p))=transverse b q-transverse b p := by
simp [rCoord_apply,latticeEmbed,transverse]; ring
have htrans := connector_transverse_displacement b X d Y S (contactSign plus) a
hS0.le (abs_contactSign plus) ⟨ha₀,ha₁⟩ p q hp hq
have hpos : 0 < contactSign plus*rCoord b (latticeEmbed (q-p)) := by
rw [hr]; linarith [htrans.1]
have habs : |rCoord b (latticeEmbed (q-p))| ≤ 3*C*s := by
have he : |rCoord b (latticeEmbed (q-p))| =
contactSign plus*rCoord b (latticeEmbed (q-p)) := by
calc
_ = |contactSign plus*rCoord b (latticeEmbed (q-p))| := by simp [abs_mul]
_ = _ := abs_of_pos hpos
rw [he,hr]
nlinarith [htrans.2]
have hg := hgap s hs0 (latticeEmbed (q-p)) (by rw [hx]; exact hdlo)
(contactSide_of_sign plus _ hpos) habs
unfold sourceSupportCost
linarith
· intro p hp q hq
simpa only [one_div,mul_comm (2⁻¹ : ℝ) s,← div_eq_mul_inv] using
connector_tube_containment b hb X d Y S s (contactSign plus) a hd0 hs0.le hS
(abs_contactSign plus) ⟨ha₀,ha₁⟩ p q hp hq
end PlanarFPP
end OAI
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