| import Mathlib |
| 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 |
|
|
