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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Bridges.SupplyIntegral
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.Bridge
open Wiener GaussianPacking PathCutoff
attribute [local irreducible] torusDistance
lemma trialPath_all_continuous (t : ℝ) :
Continuous (fun p : (Point×(Point×Point))×(Noise×Noise) ↦
trialPath t p.1.1 p.1.2.1 p.1.2.2 p.2) := by
apply ContinuousMap.continuous_of_continuous_uncurry
change Continuous (fun p : ((Point×(Point×Point))×(Noise×Noise))×Time ↦ fun j ↦
(mean p.1.1.1 p.1.1.2.2 (leftTime p.2) j+Real.sqrt (2*t)*center (p.1.2.1 j) (leftTime p.2))+
(mean p.1.1.2.1 p.1.1.2.2 (reverseTime (rightTime p.2)) j+
Real.sqrt (2*t)*center (p.1.2.2 j) (reverseTime (rightTime p.2)))-p.1.1.2.2 j)
unfold mean
have hl := leftTime_continuous
have hr := reverseTime.continuous.comp rightTime_continuous
have hc := center_continuous
fun_prop
def exposedTrialKernel (t : ℝ) : Kernel (Point×(Point×Point)) C(Time,Point) :=
((Kernel.deterministic id measurable_id).prod
(Kernel.const (Point×(Point×Point)) (law.prod law))).map
(fun p ↦ trialPath t p.1.1 p.1.2.1 p.1.2.2 p.2)
instance exposedTrialKernel_markov (t : ℝ) : IsMarkovKernel (exposedTrialKernel t) := by
unfold exposedTrialKernel
exact Kernel.IsMarkovKernel.map _ (trialPath_all_continuous t).measurable
lemma exposedTrialKernel_apply (t : ℝ) (y z u : Point) :
exposedTrialKernel t (y,z,u)=trialKernel t y z u := by
rw [exposedTrialKernel,Kernel.map_apply _ (trialPath_all_continuous t).measurable,
Kernel.prod_apply,Kernel.deterministic_apply,Kernel.const_apply,Measure.dirac_prod,
Measure.map_map (trialPath_all_continuous t).measurable
(measurable_const.prodMk measurable_id : Measurable (Prod.mk (id (y,z,u)))),
trialKernel_apply]
rfl
lemma trialCollision_joint_closed {L : Length} (r : ℝ) :
IsClosed {p : ConfigurationPath 1 L×C(Time,Point) | TrialCollision r p.1 p.2} := by
let E : Set ((ConfigurationPath 1 L×C(Time,Point))×Time) :=
{p | torusDistance L (projectPoint L (p.1.2 p.2)) (p.1.1 p.2 0) ≤ r}
have hE : IsClosed E := by
apply isClosed_le _ continuous_const
apply (torusDistance_continuous_pair L).comp (f := fun p : (ConfigurationPath 1 L×C(Time,Point))×Time ↦
(projectPoint L (p.1.2 p.2),p.1.1 p.2 0))
apply Continuous.prodMk
· change Continuous (fun p : (ConfigurationPath 1 L×C(Time,Point))×Time ↦
fun j : Fin 3 ↦ (p.1.2 p.2 j:AddCircle L.val))
fun_prop
· fun_prop
convert isClosedMap_fst_of_compactSpace E hE using 1
ext p
simp only [TrialCollision,mem_ofPred_eq,mem_image,E]
exact ⟨fun ⟨s,hs⟩ ↦ ⟨(p,s),hs,rfl⟩,fun ⟨⟨q,s⟩,hs,hq⟩ ↦ by
change q=p at hq
subst q
exact ⟨s,hs⟩⟩
lemma successExcept_joint_measurable {L : Length} {N : ℕ} (n : ℕ)
(r R C S : ℝ) (w : Time) (v : Point) (i : Fin N) :
MeasurableSet {p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) |
p.2∈SuccessExcept n r R C S w v p.1 i} := by
have hm (j : Fin N) : Measurable (fun p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) ↦
(p.1 j,p.2)) := ((measurable_pi_apply j).comp measurable_fst).prodMk measurable_snd
have hh : MeasurableSet {p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) |
∀j,j≠i→¬TrialCollision r (p.1 j) p.2} := by
simp only [ofPred_forall]
exact MeasurableSet.iInter fun j ↦ MeasurableSet.iInter fun _ ↦
((trialCollision_joint_closed r).measurableSet.preimage (hm j)).compl
exact ((trialCutoff_measurable L n r R C w S v).preimage measurable_snd).inter hh
end HardSphere.Bridge
end
end
end
end OAI
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