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import OAI.Analysis.HardSphere.Bridges.ConfinedLift
import OAI.Analysis.HardSphere.Conditioning.ConditionTransport
import OAI.Analysis.HardSphere.Allocation.AllocationJoint
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.Groups
open Wiener Bridge GaussianPacking PathCutoff Comparison CellOrbital
attribute [local instance] Classical.propDecidable
def frozenPaths {N : ℕ} {L : Length} (t : ℝ) (e : BridgeEndpoints N L)
(p : Fin N→Prop) (w : {i // ¬p i}→Bridge.Noise) : {i // ¬p i} → ConfigurationPath 1 L :=
fun j ↦ projectCurve L (leg t (labelStartLift e j) (labelEndLift t e j) (w j))
lemma actual_reference_event {N K : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time)
(e : BridgeEndpoints N L) (p : Fin N→Prop) (w : {i // ¬p i}→Bridge.Noise)
(b : Fin N→Bool) (i : {i // p i}) (hbi : b i=true)
(v : Fin 3→Fin K) (vLift : Point)
(hvLift : projectPoint L vLift=gridCenter L v)
(hy : ‖labelStartLift e i-vLift‖≤3)
(he : EndpointEligible v s₀ e.2 (bridgeEnd N L (2*t) e) i) :
individualFiber p (noisePairAvoid r (2*t) e)
(fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (e,β)) b w i
=ᵐ[Bridge.law]
(leg (2*t) (labelStartLift e i) (labelEndLift (2*t) e i))⁻¹'
permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i)
vLift (frozenPaths (2*t) e p w) := by
filter_upwards [leg_endpoints_ae (2*t) (labelStartLift e i) (labelEndLift (2*t) e i)] with β hβ
apply propext
have hown := ownEligible_at hL hK m r R C S s₀ (2*t) u i e β v he
change (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u i (e,β)=b i ∧
∀j : {j // ¬p j},noisePairAvoid r (2*t) e i j β (w j) ∧ noisePairAvoid r (2*t) e j i (w j) β) ↔ _
rw [hbi]
simp only [eligibilityFlag,decide_eq_true_eq,hown]
constructor
· rintro ⟨⟨hreg,hconf⟩,havoid⟩
refine ⟨hreg,?_,hβ,?_⟩
· apply confined_lift h50 vLift _ (by simpa only [hβ.1] using hy)
intro s
rw [hvLift]
exact hconf s
· intro j
exact (havoid j).1
· rintro ⟨hreg,hconf,_,havoid⟩
refine ⟨⟨hreg,?_⟩,?_⟩
· intro s
rw [←hvLift]
exact (hconf s).2
· intro j
refine ⟨havoid j,?_⟩
exact fun h ↦ havoid j ((physicalPairCheck_symmetric r _ _).mp h)
lemma actual_individual_reference {N K : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time)
(e : BridgeEndpoints N L) (p : Fin N→Prop) (w : {i // ¬p i}→Bridge.Noise)
(b : Fin N→Bool) (i : {i // p i}) (hbi : b i=true)
(v : Fin 3→Fin K) (vLift : Point)
(hvLift : projectPoint L vLift=gridCenter L v)
(hy : ‖labelStartLift e i-vLift‖≤3)
(he : EndpointEligible v s₀ e.2 (bridgeEnd N L (2*t) e) i) :
(conditionedMeasure Bridge.law (individualFiber p (noisePairAvoid r (2*t) e)
(fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (e,β)) b w i)).map
(leg (2*t) (labelStartLift e i) (labelEndLift (2*t) e i))=
individualReference t (labelStartLift e i) (labelEndLift (2*t) e i)
(permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i)
vLift (frozenPaths (2*t) e p w)) := by
exact noise_reference_identification t _ _ _ (permittedPath_measurable ..)
(actual_reference_event hL hK h50 m r R C S s₀ t u e p w b i hbi v vLift hvLift hy he)
end HardSphere.Groups
end
end
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.Groups
open Wiener Bridge GaussianPacking Comparison CellOrbital
attribute [local instance] Classical.propDecidable
def physicalMutual {I : Type*} {L : Length} (r : ℝ) : Set (I → C(Time,Point)) :=
pairEvent (fun _ _ f g ↦ ¬TrialCollision (L:=L) r (projectCurve L g) f)
lemma physicalMutual_measurable {I : Type*} [Fintype I] {L : Length} (r : ℝ) :
MeasurableSet (physicalMutual (I:=I) (L:=L) r) :=
pairEvent_measurable _ (fun _ _ ↦ (jointCollision_closed (L:=L) r).measurableSet.compl)
lemma physical_product_reference {N K : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time)
(e : BridgeEndpoints N L) (p : Fin N → Prop) (w : {i // ¬p i} → Bridge.Noise)
(b : Fin N → Bool) (hbi : ∀i : {i // p i},b i=true)
(v : {i // p i} → Fin 3 → Fin K) (vLift : {i // p i} → Point)
(hvLift : ∀i,projectPoint L (vLift i)=gridCenter L (v i))
(hy : ∀i : {i // p i},‖labelStartLift e i-vLift i‖≤3)
(he : ∀i : {i // p i},EndpointEligible (v i) s₀ e.2 (bridgeEnd N L (2*t) e) i)
(hnormal : ∀i : {i // p i},Bridge.law
(individualFiber p (noisePairAvoid r (2*t) e)
(fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (e,β)) b w i)≠0) :
(conditionedMeasure (Measure.pi fun i : {i // p i} ↦
conditionedMeasure Bridge.law (individualFiber p (noisePairAvoid r (2*t) e)
(fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (e,β)) b w i))
(pairEvent (fun i j : {i // p i} ↦ noisePairAvoid r (2*t) e i j))).map
(fun β (i : {i // p i}) ↦ leg (2*t) (labelStartLift e i) (labelEndLift (2*t) e i) (β i))=
conditionedMeasure (Measure.pi fun i : {i // p i} ↦
individualReference t (labelStartLift e i) (labelEndLift (2*t) e i)
(permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i)
(vLift i) (frozenPaths (2*t) e p w))) (physicalMutual (L:=L) r) := by
let μ := fun i : {i // p i} ↦ conditionedMeasure Bridge.law
(individualFiber p (noisePairAvoid r (2*t) e)
(fun j β ↦ eligibilityFlag (K:=K) m r R C S s₀ (2*t) u j (e,β)) b w i)
let : ∀i,IsProbabilityMeasure (μ i) := fun i ↦ conditioned_probability Bridge.law (hnormal i)
have hh := conditioned_pi_map μ
(fun i ↦ leg (2*t) (labelStartLift e i) (labelEndLift (2*t) e i))
(fun i ↦ (leg_continuous ..).measurable) (physicalMutual_measurable (I:={i // p i}) (L:=L) r)
have hm : ∀i,(μ i).map (leg (2*t) (labelStartLift e i) (labelEndLift (2*t) e i))=
individualReference t (labelStartLift e i) (labelEndLift (2*t) e i)
(permittedPath m r R C S u (labelStartLift e i) (labelEndLift (2*t) e i)
(vLift i) (frozenPaths (2*t) e p w)) := fun i ↦
actual_individual_reference hL hK h50 m r R C S s₀ t u e p w b i (hbi i)
(v i) (vLift i) (hvLift i) (hy i) (he i)
simp_rw [hm] at hh
exact hh
lemma physical_recorded_reference {N K : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time)
(p : Fin N → Prop) (b : Fin N → Bool) (hbi : ∀i : {i // p i},b i=true) :
∀ᵐw ∂exposedLaw μ (fun _ : Fin N ↦ Bridge.law) p
(labelInteraction N L r (2*t)∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b),
∀(v : {i // p i} → Fin 3 → Fin K) (vLift : {i // p i} → Point),
(∀i,projectPoint L (vLift i)=gridCenter L (v i)) →
(∀i : {i // p i},‖labelStartLift w.1 i-vLift i‖≤3) →
(∀i : {i // p i},EndpointEligible (v i) s₀ w.1.2 (bridgeEnd N L (2*t) w.1) i) →
(conditionedMeasure (Measure.pi fun _ : {i // p i} ↦ Bridge.law)
{β | blockGlue p (w,β)∈labelInteraction N L r (2*t)∩
flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b}).map
(fun β (i : {i // p i}) ↦ leg (2*t) (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i) (β i))=
conditionedMeasure (Measure.pi fun i : {i // p i} ↦
individualReference t (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i)
(permittedPath m r R C S u (labelStartLift w.1 i) (labelEndLift (2*t) w.1 i)
(vLift i) (frozenPaths (2*t) w.1 p w.2))) (physicalMutual (L:=L) r) := by
filter_upwards [recorded_physical_fiber μ r (2*t)
(eligibilityFlag (K:=K) m r R C S s₀ (2*t) u)
(fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b p] with w hw
intro v vLift hv hy he
rw [hw.2.2]
exact physical_product_reference hL hK h50 m r R C S s₀ t u w.1 p w.2 b hbi v vLift hv hy he hw.2.1
end HardSphere.Groups
end
end
end
end OAI
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