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import OAI.Analysis.HardSphere.Allocation.PhysicalCellUniform
import OAI.Analysis.HardSphere.Allocation.PhysicalReferenceRaw
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Comparison
variable {Ω A G Y J : Type*} [MeasurableSpace Ω] [MeasurableSpace Y]
[Fintype A] [DecidableEq A] [Fintype G] [DecidableEq G] [Fintype J]
omit [DecidableEq A] [Fintype G] [DecidableEq G] in
lemma allocatedTagLaw_sum (μ : J→Measure Ω) (I : G→Ω→Finset A)
(hI : ∀g i,MeasurableSet {ω | i∈I g ω}) (κ : A→Kernel Ω Y) (g : G) :
allocatedTagLaw (∑j,μ j) I κ g=∑j,allocatedTagLaw (μ j) I κ g := by
classical
apply Measure.ext
intro s hs
simp only [allocatedTagLaw,Measure.finsetSum_apply]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i _
simp only [Measure.bind_apply hs (κ i).aemeasurable,←lintegral_indicator (hI g i)]
exact lintegral_finsetSum_measure _ _ _
end HardSphere.Comparison
end
end
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
attribute [local instance] Classical.propDecidable
attribute [local instance] Comparison.permBathSpace Comparison.permBathSingleton
def physicalRead {N K M G : ℕ} (L : Length) (extra tag : Fin 3→Fin K) (b : Bool)
(i : Fin (N+1)) : Kernel (ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M))
(Torus L×Torus L×Configuration N L) :=
((physicalCellPost L extra tag b) ∘ₖ orderedKernel (uniformBathOrder N) i).comap
(fun p ↦ p.1 midpointTime) ((continuous_eval_const midpointTime).measurable.comp measurable_fst)
lemma physical_branch_allocated_reference {N K M G : ℕ} [NeZero K] {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(μ : Measure (BridgeEndpoints (N+1) L)) [SFinite μ]
(g : Fin G) (flags : Fin (N+1)→Bool) (a : GridAllocation (N+1) K (G*M))
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time) (extra tag : Fin 3→Fin K) (b : Bool) :
taggedPhysicalBathKernel (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) b ∘ₘ
taggedReference
((physicalBranchLaw hL μ g flags a m r R C S s₀ (2*t) u).compProd
(physicalPrimaryReference hL g m t r R C S u
(fun i : {i // i∈groupUnion g a} ↦ i.val) extra))
(physicalTagEvent (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) tag) =
allocatedTagLaw ((rawPhysicalBranch hL μ g flags a m r R C S s₀ (2*t) u).map
(fun z ↦ (labelBridgePath (N+1) L (2*t) z,a)))
(fun g p ↦ groupUnion g p.2) (physicalRead L extra tag b) g := by
classical
rw [physical_branch_observed_reference hL hK h50]
unfold allocatedTagLaw
apply Finset.sum_congr rfl
intro i _
have hmap : Measurable (fun z ↦ (labelBridgePath (N+1) L (2*t) z,a)) :=
(labelBridgePath_measurable (N+1) L (2*t)).prodMk measurable_const
have hA : MeasurableSet {p : ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M) | i∈groupUnion g p.2} :=
((measurable_of_countable (fun a ↦ i∈groupUnion g a)).comp measurable_snd).setOf
rw [Measure.restrict_map hmap hA]
by_cases hi : i∈groupUnion g a
· rw [ite_eq_left hi]
have he : (fun z ↦ (labelBridgePath (N+1) L (2*t) z,a)) ⁻¹'
{p : ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M) | i∈groupUnion g p.2}=univ := by
ext z; simp [hi]
rw [he,Measure.restrict_univ,physicalRead,comp_comap_map,
Measure.map_map (show Measurable (fun p : ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M) ↦ p.1 midpointTime) from
(continuous_eval_const midpointTime).measurable.comp measurable_fst) hmap]
rfl
· rw [ite_eq_right hi]
have he : (fun z ↦ (labelBridgePath (N+1) L (2*t) z,a)) ⁻¹'
{p : ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M) | i∈groupUnion g p.2}=∅ := by
ext z; simp [hi]
simp [he]
theorem physical_branch_reference_sum {N K M G : ℕ} [NeZero K] {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(ψ : Jet (N+1) L) (g : Fin G) (m : ℕ) (r R C S s₀ t : ℝ) (u : Time)
(extra tag : Fin 3→Fin K) (b : Bool) :
(∑flags,∑a : GridAllocation (N+1) K (G*M),
taggedPhysicalBathKernel (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) b ∘ₘ
taggedReference
((physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g flags a m r R C S s₀ (2*t) u).compProd
(physicalPrimaryReference hL g m t r R C S u
(fun i : {i // i∈groupUnion g a} ↦ i.val) extra))
(physicalTagEvent (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) tag)) =
allocatedTagLaw ((groundPathLaw (N+1) L r (2*t) ψ).compProd
(physicalAllocationKernel m (G*M) r R C S s₀ u))
(fun g p ↦ groupUnion g p.2) (physicalRead L extra tag b) g := by
have hI : ∀g : Fin G,∀i : Fin (N+1),MeasurableSet
{p : ConfigurationPath (N+1) L×GridAllocation (N+1) K (G*M) | i∈groupUnion g p.2} := by
intro g i
exact ((measurable_of_countable (fun a ↦ i∈groupUnion g a)).comp measurable_snd).setOf
simp_rw [physical_branch_allocated_reference hL hK h50]
simp_rw [←allocatedTagLaw_sum _ _ hI]
congr 1
exact genuine_ground_allocated_raw hL hK ψ m (G*M) r R C S s₀ (2*t) u
theorem physical_allocated_reference_le {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ}
(hL : L.val=(K:ℝ)) (hK : 1≤K) {ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ)
(m : ℕ) (R C S s₀ t : ℝ) (ht : 0≤t) (u : Time)
(extra tag : Fin 3→Fin K) (b : Bool) :
(∑g : Fin G,allocatedTagLaw ((groundPathLaw (N+1) L r (2*t) ψ).compProd
(physicalAllocationKernel m (G*M) r R C S s₀ u))
(fun g p ↦ groupUnion g p.2) (physicalRead L extra tag b) g) ≤
(N+1) • (physicalCellPost L extra tag b ∘ₘ
(((volume : Measure (Torus L)).prod volume).withDensity
(fun p : Torus L×Configuration N L ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2)))) := by
have hT : 0≤2*t := mul_nonneg (by norm_num) ht
let := hψ.groundPathLaw_probability hT
refine (sum_physical_group_tags_le hL hK m r R C S s₀ u _ _).trans ?_
simp_rw [physicalRead,comp_comap_map,allocation_slice_law hψ (2*t) hT midpointTime]
exact le_of_eq (all_tags_reference ψ hψ.2.1 hψ.2.2.1 (uniformBathOrder N) (physicalCellPost L extra tag b))
end HardSphere.Groups
end
end
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal BigOperators
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
def physicalCellVolumePost {N K : ℕ} (L : Length) (extra tag : Fin 3→Fin K) (b : Bool) :
Kernel (Torus L×Configuration N L) (Torus L×Torus L×Configuration N L) :=
(cellExtraKernel volume (centeredCell L (gridCenter L tag))
(centeredCell L (gridCenter L extra)) (centeredCell_measurable L _)).map (cellObservation b)
theorem physicalCellPost_le {N K : ℕ} [NeZero K] {L : Length} (hL : 1≤L.val)
(extra tag : Fin 3→Fin K) (b : Bool) (P : Measure (Torus L×Configuration N L)) [SFinite P] :
physicalCellPost L extra tag b ∘ₘ P≤physicalCellVolumePost L extra tag b ∘ₘ P := by
rw [physicalCellPost,physicalCellVolumePost,←Measure.map_comp _ _ (cellObservation_measurable b),
←Measure.map_comp _ _ (cellObservation_measurable b)]
apply Measure.map_mono _ (cellObservation_measurable b)
rw [cellExtraKernel,cellExtraKernel,restrict_kernel_comp,restrict_kernel_comp,
←Measure.compProd_eq_comp_prod,←Measure.compProd_eq_comp_prod,
Measure.compProd_const,Measure.compProd_const,Measure.restrict_univ]
apply Measure.restrict_mono_measure
exact Measure.prod_mono le_rfl (cellPathReference_midpoint_le hL extra)
theorem physical_all_references_le {N K M G : ℕ} [NeZero K] {L : Length} {r : ℝ}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
{ψ : Jet (N+1) L} (hψ : IsGroundVector (N+1) L r ψ)
(m : ℕ) (R C S s₀ t : ℝ) (ht : 0≤t) (u : Time)
(extra tag : Fin 3→Fin K) (b : Bool) :
(∑g : Fin G,∑flags,∑a : GridAllocation (N+1) K (G*M),
taggedPhysicalBathKernel (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) b ∘ₘ
taggedReference
((physicalBranchLaw hL (endpointWeightedBase ψ (2*t)) g flags a m r R C S s₀ (2*t) u).compProd
(physicalPrimaryReference hL g m t r R C S u
(fun i : {i // i∈groupUnion g a} ↦ i.val) extra))
(physicalTagEvent (Function.Embedding.subtype (fun i ↦ i∈groupUnion g a)) tag)) ≤
(N+1) • (physicalCellVolumePost L extra tag b ∘ₘ
(((volume : Measure (Torus L)).prod volume).withDensity
(fun p : Torus L×Configuration N L ↦ ENNReal.ofReal (‖ψ.1 (Fin.cons p.1 p.2)‖^2)))) := by
simp_rw [physical_branch_reference_sum hL hK h50]
exact (physical_allocated_reference_le hL hK hψ m R C S s₀ t ht u extra tag b).trans
(nsmul_le_nsmul_right (physicalCellPost_le (by linarith) extra tag b _) (N+1))
end HardSphere.Groups
end
end
end OAI
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