| import Mathlib | |
| 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 | |