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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 | import OAI.Probability.DirectionalWalk.ExperimentContact
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
def latePrefixEvent {d ι : ℕ} (e : Step d) (k r N : ℕ)
(E : Word d → Fin ι → Set (Path d)) (start : Word d → Fin ι → Site d) :
Set (Path d × Path d) :=
⋃ a : {a : Word d | AxisBridgeWord e k a}, ⋃ q : Fin ι,
⋃ b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b},
(wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) ×ˢ
(shiftPath (start a q) ⁻¹' contactAvoid b)
lemma latePrefixEvent_bound {d ι : ℕ} [NeZero ι]
(μ : Measure (Row d)) [IsProbabilityMeasure μ] (hell : StrictEllipticity μ)
(e : Step d) (k r N m : ℕ) (hrN : r ≤ N)
(E : Word d → Fin ι → Set (Path d)) (hE : ∀ a q, MeasurableSet (E a q))
(hdis : ∀ a, Pairwise (fun i j => Disjoint (E a i) (E a j)))
(start : Word d → Fin ι → Site d)
(hstart : ∀ a, AxisBridgeWord e k a → ∀ q, axisHeight e (start a q) = (N : ℤ)-1)
(hΔ : axisReachProb μ e r-axisReachProb μ e N ≤ ENNReal.ofReal ((1/2 : ℝ)^m)) :
((annealed μ 0).prod (conditioned μ (axisDirection (oppositeStep e))))
(latePrefixEvent e k r N E start) ≤
(annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ *
ENNReal.ofReal ((24 / posteriorExponent)*Real.log (ι+1)*(m+1)^2*(1/2 : ℝ)^m) := by
have : IsFiniteMeasure (conditioned μ (axisDirection (oppositeStep e))) := by
unfold conditioned
infer_instance
unfold latePrefixEvent
calc
_ ≤ ∑' a : {a : Word d | AxisBridgeWord e k a}, ∑' q : Fin ι,
∑' b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b},
((annealed μ 0).prod (conditioned μ (axisDirection (oppositeStep e))))
((wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) ×ˢ
(shiftPath (start a q) ⁻¹' contactAvoid b)) := by
apply (measure_iUnion_le _).trans
apply ENNReal.tsum_le_tsum
intro a
apply (measure_iUnion_le _).trans
exact ENNReal.tsum_le_tsum (fun q => measure_iUnion_le _)
_ = (annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ *
(∑' a : {a : Word d | AxisBridgeWord e k a}, ∑ q,
∑' b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b},
annealed μ 0 (wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) *
annealed μ (start a q) (avoids (axisUpper e N) ∩ contactAvoid b)) := by
rw [← ENNReal.tsum_mul_left]
apply tsum_congr
intro a
rw [tsum_fintype,Finset.mul_sum]
apply Finset.sum_congr rfl
intro q _
rw [← ENNReal.tsum_mul_left]
apply tsum_congr
intro b
rw [Measure.prod_prod,conditioned_negative_shift μ e _ N (hstart _ a.property _) _ (measurableSet_contactAvoid _)]
ac_rfl
_ ≤ _ := mul_le_mul_right (prefix_contact_total_bound μ hell e k r N m hrN E hE hdis start hΔ) _
def prependRelative {d : ℕ} (a : Word d) (R : Path d) : Path d :=
concatPath a.1 (wordPath a) (shiftPath (wordEnd a) R)
lemma measurable_prependRelative {d : ℕ} (a : Word d) : Measurable (prependRelative a) := by
apply Measurable.of_eval
intro j
by_cases hj : j < a.1
· simpa only [prependRelative,concatPath,ite_eq_left hj] using
(measurable_const : Measurable (fun _ : Path d => wordPath a j))
· have hm : Measurable (fun R : Path d => R (j-a.1)) := measurable_pi_apply _
simpa only [prependRelative,concatPath,ite_eq_right hj,shiftPath] using hm.add_const (wordEnd a)
lemma prependRelative_recover {d : ℕ} (a : Word d) (X : Path d) (hX : X ∈ wordCylinder a) :
prependRelative a (shiftPath (-wordEnd a) (tailPath a.1 X)) = X := by
funext j
by_cases hj : j < a.1
· simpa only [prependRelative,concatPath,ite_eq_left hj] using (hX j hj.le).symm
· simp only [prependRelative,concatPath,ite_eq_right hj,shiftPath,tailPath,
Nat.add_sub_of_le (Nat.le_of_not_gt hj),neg_add_cancel_right]
lemma latePrefixEvent_of_contact {d ι : ℕ} (e : Step d)
(k r N : ℕ) (hr : k ≤ r) (hN : k ≤ N)
(E : Fin ι → Set (Path d)) (z : Fin ι → Site d)
(X Y : Path d) (q : Fin ι) (hE : X ∈ E q)
(t τ : ℕ) (htτ : t ≤ τ)
(hold : AxisBridgeWord e k (prefixWord t X))
(hband : ∀ j, t ≤ j → j ≤ τ → (k : ℤ) ≤ axisHeight e (X j) ∧ axisHeight e (X j) < (N : ℤ))
(hreached : ∃ j, t ≤ j ∧ j ≤ τ ∧ (r : ℤ) ≤ axisHeight e (X j))
(hhit : ∃ j, Y j+z q = X τ)
(hbefore : ∀ i, t ≤ i → i < τ → ∀ j, Y j+z q ≠ X i) :
(X,Y) ∈ latePrefixEvent e k (r-k) (N-k)
(fun a q => prependRelative a ⁻¹' E q) (fun a q => z q-wordEnd a) := by
let a := prefixWord t X
let R := shiftPath (-wordEnd a) (tailPath t X)
let b := prefixWord (τ-t) R
have ha : X ∈ wordCylinder a := fun j hj => (wordPath_prefixWord t X hj).symm
have hend : wordEnd a = X t := wordPath_prefixWord t X le_rfl
have hbpath (j : ℕ) (hj : j ≤ τ-t) : wordPath b j = X (t+j)-X t := by
rw [wordPath_prefixWord (τ-t) R hj]
change X (t+j)+(-wordEnd a) = _
rw [hend,sub_eq_add_neg]
have hb : upperContactWord 0 (axisUpper e ((r-k : ℕ) : ℤ))
(axisLower e 0 ∪ axisUpper e ((N-k : ℕ) : ℤ)) b := by
refine ⟨?_,?_,?_⟩
· rw [hbpath 0 (Nat.zero_le _),Nat.add_zero,sub_self]
· obtain ⟨j,htj,hjτ,hjr⟩ := hreached
refine ⟨j-t,by change j-t ≤ τ-t;omega,?_⟩
rw [hbpath (j-t) (by omega),Nat.add_sub_of_le htj]
change ((r-k : ℕ) : ℤ) ≤ axisHeight e (X j-X t)
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,← hend,hold.1,Nat.cast_sub hr]
omega
· intro j hj
rw [hbpath j hj]
have hjτ : t+j ≤ τ := by change j ≤ τ-t at hj;omega
obtain ⟨hlo,hhi⟩ := hband (t+j) (by omega) hjτ
change ¬(axisHeight e (X (t+j)-X t) < (0 : ℤ) ∨
((N-k : ℕ) : ℤ) ≤ axisHeight e (X (t+j)-X t))
rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,← hend,hold.1,Nat.cast_sub hN]
omega
have hca : shiftPath (z q-wordEnd a) Y ∈ contactAvoid b := by
rw [contactAvoid_iff_firstHit]
constructor
· obtain ⟨j,hj⟩ := hhit
refine ⟨j,?_⟩
change Y j+(z q-wordEnd a) = wordPath b (τ-t)
rw [hbpath (τ-t) le_rfl,Nat.add_sub_of_le htτ,hend]
rw [sub_eq_add_neg,← add_assoc,hj,sub_eq_add_neg]
· intro i hi
rintro ⟨j,hj⟩
apply hbefore (t+i) (by omega) (by change i < τ-t at hi;omega) j
rw [hbpath i hi.le] at hj
change Y j+(z q-wordEnd a) = X (t+i)-X t at hj
rw [hend,sub_eq_add_neg,sub_eq_add_neg,← add_assoc] at hj
exact add_right_cancel hj
refine Set.mem_iUnion.mpr ⟨⟨a,hold⟩,Set.mem_iUnion.mpr ⟨q,Set.mem_iUnion.mpr ⟨⟨b,hb⟩,?_⟩⟩⟩
refine ⟨⟨ha,?_,?_⟩,hca⟩
· change prependRelative a R ∈ E q
exact (prependRelative_recover a X ha).symm ▸ hE
· exact fun j hj => (wordPath_prefixWord (τ-t) R hj).symm
end DirectionalZeroOne
end OAI
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