| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalCodedMoment |
| import OAI.Analysis.HardSphere.Comparison.PhysicalObservedLaw |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital _root_.OAI.HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
|
|
| def physicalCodeTag {N : ℕ} (particle : J↪Fin (N+1)) (hxy : x≠y) |
| (b : Bool) (c : PhysicalRouteCode x y J) : Fin (N+1) := |
| if h : RouteFieldValid c.1 c.2 then |
| particle (fieldRouteLabels hxy c.1 c.2 h (if b then 0 else Fin.last (c.1.val.length-1))) |
| else 0 |
|
|
| def taggedPhysicalBathKernel {N M G : ℕ} {L : Length} (particle : J↪Fin (N+1)) (b : Bool) : |
| Kernel (Fin (N+1)×(PhysicalDatum (N+1) K M G L×(Option J→C(Time,Point)))) |
| (Torus L×Torus L×(Fin N→Torus L)) where |
| toFun z := primaryBathKernel b z.1 (physicalFullMidpoint particle z.2) |
| measurable' := measurable_from_prod_countable_right fun i ↦ |
| (primaryBathKernel b i).measurable.comp (physicalFullMidpoint_measurable particle) |
| instance taggedPhysicalBathKernel_markov {N M G : ℕ} {L : Length} (particle : J↪Fin (N+1)) (b : Bool) : |
| IsMarkovKernel (taggedPhysicalBathKernel (K:=K) (M:=M) (G:=G) (L:=L) particle b) := |
| ⟨fun z ↦ inferInstanceAs (IsProbabilityMeasure (primaryBathKernel b z.1 (physicalFullMidpoint particle z.2)))⟩ |
| |
| def physicalCodeObserver {N M G : ℕ} {L : Length} (particle : J↪Fin (N+1)) |
| (hxy : x≠y) (b : Bool) : |
| Kernel (PhysicalDatum (N+1) K M G L×(PhysicalRouteCode x y J×((Option J→C(Time,Point))×(Option J→C(Time,Point))))) |
| (Torus L×Torus L×(Fin N→Torus L)) := |
| (taggedPhysicalBathKernel particle b).comap |
| (fun z ↦ (physicalCodeTag particle hxy b z.2.1,(z.1,if b then z.2.2.1 else z.2.2.2))) |
| (by |
| apply Measurable.prodMk |
| · exact (measurable_of_countable (physicalCodeTag particle hxy b)).comp measurable_snd.fst |
| · cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true] <;> exact measurable_fst.prodMk (by fun_prop)) |
| instance physicalCodeObserver_markov {N M G : ℕ} {L : Length} (particle : J↪Fin (N+1)) |
| (hxy : x≠y) (b : Bool) : |
| IsMarkovKernel (physicalCodeObserver (M:=M) (G:=G) (L:=L) particle hxy b) := by |
| unfold physicalCodeObserver; infer_instance |
| |
| theorem physicalCodedRoute_same_observed {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (u : Time) |
| (particle : J↪Fin (N+1)) (d : PhysicalDatum (N+1) K M G L) (hxy : x≠y) |
| (p : SimplePath x y) (z : DiscreteTorus K→Option J) |
| (hz : ∀v,z v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d)) |
| (ha : ∀i : Fin p.val.length,(branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) : |
| ((physicalCodeObserver particle hxy true).comap (fun w ↦ (d,((p,z),w))) |
| (measurable_const.prodMk (measurable_const.prodMk measurable_id))) ∘ₘ |
| physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d = |
| ((physicalCodeObserver particle hxy false).comap (fun w ↦ (d,((p,z),w))) |
| (measurable_const.prodMk (measurable_const.prodMk measurable_id))) ∘ₘ |
| physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d := by |
| have hv := routeFieldValid_of_supported hL g m t r R C S u particle d p z hz ha |
| have hh := fieldRouteLabels_physical hL g m t r R C S u particle d hxy p z hz ha hv |
| have H := physicalPairedRoute_same_observed hL g m t r R C S ht u particle |
| (fieldRouteLabels hxy p z hv) (simpleRouteVertices p (p.val.length-1)) |
| (simpleRouteSteps hL p _ (simpleRouteLength_eq hxy p)) |
| (simpleRouteSteps_period hL p _ (simpleRouteLength_eq hxy p)) d |
| (fun i ↦ (hh i).1) (fun i ↦ (hh i).2) |
| simp only [physicalCodedRoute,dite_eq_left hv] |
| convert H using 2 <;> |
| (funext w |
| simp only [physicalCodeObserver,Kernel.comap_apply,taggedPhysicalBathKernel,Kernel.coe_mk, |
| physicalCodeTag,dite_eq_left hv,ite_true,Bool.false_eq_true,ite_false,physicalBathKernel]) |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital _root_.OAI.HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} [NeZero K] {x y : DiscreteTorus K} |
| variable {J : Type*} [Fintype J] [DecidableEq J] |
| |
| omit [NeZero K] in |
| lemma unitTarget_project {L : Length} (hL : 1≤L.val) (v : Fin 3→Fin K) (z : Point) |
| (hz : z∈unitTarget (anchorPoint v)) : projectPoint L z∈centeredCell L (gridCenter L v) := by |
| rw [centeredCell_image hL] |
| intro j |
| have hj := hz j (mem_univ j) |
| refine ⟨z j-(v j:ℝ),⟨?_,?_⟩,?_⟩ |
| · change (v j:ℝ)-1/2≤z j ∧ z j<(v j:ℝ)+1/2 at hj |
| linarith [hj.1] |
| · change (v j:ℝ)-1/2≤z j ∧ z j<(v j:ℝ)+1/2 at hj |
| linarith [hj.2] |
| · change ((z j-(v j:ℝ):ℝ):AddCircle L.val)=((z j:ℝ):AddCircle L.val)-((v j:ℝ):AddCircle L.val) |
| exact QuotientAddGroup.mk_sub _ _ _ |
| |
| omit [NeZero K] in |
| lemma cellPathReference_support {L : Length} (hL : 1≤L.val) (v : Fin 3→Fin K) : |
| ∀ᵐp ∂cellPathReference v,projectPoint L (p midpointTime)∈centeredCell L (gridCenter L v) := by |
| have hm : Measurable (fun p : C(Time,Point) ↦ projectPoint L (p midpointTime)) := |
| (projectPoint_continuous L).measurable.comp (continuous_eval_const midpointTime).measurable |
| rw [cellPathReference,ae_map_iff pointAsPath_measurable.aemeasurable ((centeredCell_measurable L _).preimage hm)] |
| unfold targetUniform |
| apply Measure.ae_smul_measure |
| filter_upwards [ae_restrict_mem (unitTarget_measurable (anchorPoint v))] with z hz |
| exact unitTarget_project hL v z hz |
| |
| omit [DecidableEq J] in |
| lemma physicalPrimary_extra {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (particle : J↪Fin N) |
| (v : Fin 3→Fin K) (d : PhysicalDatum N K M G L) : |
| ∀ᵐw ∂physicalPrimaryReference hL g m t r R C S u particle v d, |
| physicalFullMidpoint particle (d,w) none∈centeredCell L (gridCenter L v) := by |
| have hL1 : 1≤L.val := by rw [hL]; exact_mod_cast NeZero.pos K |
| have hextra := (Measure.quasiMeasurePreserving_eval |
| (fun i ↦ physicalSlotReference hL g m t r R C S u particle v i d) none).ae (cellPathReference_support hL1 v) |
| rw [physicalPrimaryReference_apply,conditionedMeasure] |
| apply Measure.ae_smul_measure |
| filter_upwards [ae_restrict_of_ae hextra] with w hw |
| change fillGroup particle _ _ (optionParticle particle none)∈_ |
| rw [fillGroup_selected] |
| exact hw |
| |
| def physicalTagEvent {N M G : ℕ} {L : Length} (particle : J↪Fin N) |
| (v : Fin 3→Fin K) (i : Fin N) : Set (PhysicalDatum N K M G L×(Option J→C(Time,Point))) := |
| {z | i∈Set.range particle ∧ physicalFullMidpoint particle z (some i)∈centeredCell L (gridCenter L v)} |
| omit [NeZero K] [DecidableEq J] in |
| lemma physicalTagEvent_measurable {N M G : ℕ} {L : Length} (particle : J↪Fin N) |
| (v : Fin 3→Fin K) (i : Fin N) : MeasurableSet (physicalTagEvent (M:=M) (G:=G) (L:=L) particle v i) := |
| .inter (show MeasurableSet {z : PhysicalDatum N K M G L×(Option J→C(Time,Point)) | i∈Set.range particle} from by |
| by_cases h : i∈Set.range particle <;> simp [h]) ((centeredCell_measurable L _).preimage |
| ((measurable_pi_apply (some i)).comp (physicalFullMidpoint_measurable particle))) |
| |
| theorem physicalCodedRoute_tag_support {N M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (hr : r≤1) (u : Time) |
| (particle : J↪Fin (N+1)) (d : PhysicalDatum (N+1) K M G L) (hxy : x≠y) |
| (p : SimplePath x y) (z : DiscreteTorus K→Option J) |
| (hz : ∀v,z v∈paddedLabels (branchAccessible hL g m t r R C S u particle v d)) |
| (ha : ∀i : Fin p.val.length,(branchAccessible hL g m t r R C S u particle (pathVertexEmbedding p i.castSucc) d).Nonempty) |
| (hp : ∀j,IsProbabilityMeasure (datumIndividualReference hL g m r R C S t u (particle j) d)) (b : Bool) : |
| ∀ᵐw ∂physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d, |
| (d,if b then w.1 else w.2)∈physicalTagEvent particle (siteIndex (if b then x else y)) |
| (physicalCodeTag particle hxy b (p,z)) := by |
| have he (b : Bool) : ∀ᵐw ∂physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d, |
| physicalFullMidpoint particle (d,if b then w.1 else w.2) none∈ |
| centeredCell L (gridCenter L (siteIndex (if b then y else x))) := by |
| have hd := physicalCoded_density hL g m t r R C S ht hr u particle d hxy p z hz ha hp b |
| have h := (withDensity_absolutelyContinuous _ (physicalRouteLikelihood hL g m t r R C S u particle hxy b (d,(p,z)))).ae_le (physicalPrimary_extra hL g m t r R C S u particle (siteIndex (if b then y else x)) d) |
| rw [hd] at h |
| have hf : Measurable (if b then (Prod.fst : ((Option J→C(Time,Point))×(Option J→C(Time,Point)))→_) else Prod.snd) := by cases b <;> simp <;> fun_prop |
| have hm : Measurable (fun w : Option J→C(Time,Point) ↦ physicalFullMidpoint particle (d,w) none) := |
| (measurable_pi_apply none).comp ((physicalFullMidpoint_measurable particle).comp measurable_prodMk_left) |
| have h' := (ae_map_iff hf.aemeasurable ((centeredCell_measurable L _).preimage hm)).mp h |
| cases b <;> simpa using h' |
| have hv := routeFieldValid_of_supported hL g m t r R C S u particle d p z hz ha |
| have hh := fieldRouteLabels_physical hL g m t r R C S u particle d hxy p z hz ha hv |
| have hmid := physicalPairedRoute_midpoint hL g m t r R C S ht u particle |
| (fieldRouteLabels hxy p z hv) (simpleRouteVertices p (p.val.length-1)) |
| (simpleRouteSteps hL p _ (simpleRouteLength_eq hxy p)) |
| (simpleRouteSteps_period hL p _ (simpleRouteLength_eq hxy p)) d |
| (fun i ↦ (hh i).1) (fun i ↦ (hh i).2) |
| have hmid' : ∀ᵐw ∂physicalCodedRoute hL g m t r R C S u particle hxy (p,z) d, |
| ∀i,physicalFullMidpoint particle (d,w.2) i=physicalFullMidpoint particle (d,w.1) |
| (routePermutation ((fieldRouteLabels hxy p z hv).trans particle) i) := by |
| simpa only [physicalCodedRoute,dite_eq_left hv] using hmid |
| filter_upwards [he true,he false,hmid'] with w h1 h2 hm |
| cases b with |
| | true => |
| simp only [physicalTagEvent,mem_ofPred_eq,physicalCodeTag,dite_eq_left hv,ite_true] |
| refine ⟨⟨_,rfl⟩,?_⟩ |
| have H := hm none |
| rw [routePermutation_star] at H |
| simp only [Function.Embedding.trans_apply] at H |
| change physicalFullMidpoint particle (d,w.1) (some (particle (fieldRouteLabels hxy p z hv 0)))∈_ |
| rw [←H] |
| exact h2 |
| | false => |
| simp only [physicalTagEvent,mem_ofPred_eq,physicalCodeTag,dite_eq_left hv,Bool.false_eq_true,ite_false] |
| refine ⟨⟨_,rfl⟩,?_⟩ |
| have H := hm (some (particle (fieldRouteLabels hxy p z hv (Fin.last (p.val.length-1))))) |
| change physicalFullMidpoint particle (d,w.2) (some (((fieldRouteLabels hxy p z hv).trans particle) (Fin.last (p.val.length-1)))) = physicalFullMidpoint particle (d,w.1) (routePermutation ((fieldRouteLabels hxy p z hv).trans particle) (some (((fieldRouteLabels hxy p z hv).trans particle) (Fin.last (p.val.length-1))))) at H |
| rw [routePermutation_lastLabel] at H |
| simp only [Function.Embedding.trans_apply] at H |
| rw [H] |
| exact h1 |
| |
| end HardSphere.Groups |
| end |
| end |
| |
| end OAI |
| |
