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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import Mathlib
import OAI.Analysis.HardSphere.Allocation.PhysicalBranchLaw
namespace OAI
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
attribute [local instance] Classical.propDecidable
def rawPhysicalBranch {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(_group : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M))
(m : ℕ) (r R C S s₀ T : ℝ) (u : Time) :=
((allocationBase μ (gridAllocationKernel hL (G*M)) b a).prod
(Measure.pi fun _ : Fin N ↦ Bridge.law)).restrict
(labelInteraction N L r T∩flagEvent (eligibilityFlag (K:=K) m r R C S s₀ T u) b)
instance rawPhysicalBranch_sfinite {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M))
(m : ℕ) (r R C S s₀ T : ℝ) (u : Time) :
SFinite (rawPhysicalBranch hL μ g b a m r R C S s₀ T u) := by
unfold rawPhysicalBranch; infer_instance
theorem physicalBranch_reference_raw {N K M G : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M))
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time) :
(physicalBranchLaw hL μ g b a m r R C S s₀ (2*t) u).compProd
(datumPhysicalReference hL g a m r R C S t u)=
(rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map
(fun z ↦ (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a),
fun i : {i // i∈groupUnion g a} ↦ onePathAt i (labelBridgePath N L (2*t) z))) := by
let p := fun i ↦ i∈groupUnion g a
let B := labelInteraction N L r (2*t)∩
flagEvent (eligibilityFlag (K:=K) m r R C S s₀ (2*t) u) b
have hB : MeasurableSet B := (labelInteraction_measurable N L r (2*t)).inter
(flagEvent_measurable _ (fun i ↦ eligibilityFlag_measurable m r R C S s₀ (2*t) u i) b)
have hm : Measurable (fun z : BridgeEndpoints N L×(Fin N→Bridge.Noise) ↦
(physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a),
fun i : {i // p i} ↦ onePathAt i (labelBridgePath N L (2*t) z))) :=
((physicalDatum_measurable g m r R C S s₀ u).comp
((labelBridgePath_measurable N L (2*t)).prodMk measurable_const)).prodMk
(Measurable.of_eval fun i ↦ (onePathAt_continuous (i:Fin N)).measurable.comp
(labelBridgePath_measurable N L (2*t)))
rw [physicalBranchLaw,←actual_physical_branch_joint hL hK h50 μ m r R C S s₀ t u g b a]
have : SFinite (exposedLaw (allocationBase μ (gridAllocationKernel hL (G*M)) b a)
(fun _ : Fin N ↦ Bridge.law) p B) := by unfold exposedLaw; infer_instance
have H := congrArg (fun ν ↦ ν.map (fun z ↦
(physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a),
fun i : {i // p i} ↦ onePathAt i (labelBridgePath N L (2*t) z))))
(exposed_block_reconstruct (allocationBase μ (gridAllocationKernel hL (G*M)) b a)
(fun _ : Fin N ↦ Bridge.law) p hB)
rw [Measure.map_map hm (blockGlue_measurable p)] at H
dsimp only [Function.comp_def,rawPhysicalBranch,selectedPhysicalPath,p,B] at H ⊢
convert H using 1
congr 20
all_goals try exact Subsingleton.elim _ _
funext z
unfold selectedPhysicalPath
congr 20
all_goals first | exact Subsingleton.elim _ _ | (funext index; congr 20)
def physicalRecoveredPath {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M))
(z : PhysicalDatum N K M G L×({i // i∈groupUnion g a}→ ConfigurationPath 1 L)) :
Fin N→ ConfigurationPath 1 L :=
fun i ↦ if hi : i∈groupUnion g a then z.2 ⟨i,hi⟩ else z.1.2.2.2.2 i
lemma physicalRecoveredPath_measurable {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M)) :
Measurable (physicalRecoveredPath (L:=L) g a) := by
apply Measurable.of_eval
intro i
unfold physicalRecoveredPath
by_cases hi : i∈groupUnion g a
· simp only [dite_eq_left hi]
exact (measurable_pi_apply (⟨i,hi⟩ : {i // i∈groupUnion g a})).comp measurable_snd
· simp only [dite_eq_right hi]
exact (datumFrozen_measurable i).comp measurable_fst
lemma physicalRecoveredPath_actual {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M)) (m : ℕ) (r R C S s₀ : ℝ) (u : Time)
(F : ConfigurationPath N L) :
physicalRecoveredPath g a (physicalDatum g m r R C S s₀ u (F,a),fun i ↦ onePathAt i F)=
fun i ↦ onePathAt i F := by
funext i
by_cases hi : i∈groupUnion g a <;> simp [physicalRecoveredPath,physicalDatum,hi]
def physicalRecoveredMidpoint {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M))
(z : PhysicalDatum N K M G L×({i // i∈groupUnion g a}→ ConfigurationPath 1 L)) : Configuration N L :=
fun i ↦ physicalRecoveredPath g a z i midpointTime 0
lemma physicalRecoveredMidpoint_measurable {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M)) : Measurable (physicalRecoveredMidpoint (L:=L) g a) :=
Measurable.of_eval fun i ↦ (measurable_pi_apply 0).comp
((continuous_eval_const midpointTime).measurable.comp
((measurable_pi_apply i).comp (physicalRecoveredPath_measurable g a)))
lemma physicalRecoveredMidpoint_actual {N K M G : ℕ} {L : Length} (g : Fin G)
(a : GridAllocation N K (G*M)) (m : ℕ) (r R C S s₀ : ℝ) (u : Time)
(F : ConfigurationPath N L) :
physicalRecoveredMidpoint g a (physicalDatum g m r R C S s₀ u (F,a),fun i ↦ onePathAt i F)=
F midpointTime := by
unfold physicalRecoveredMidpoint
rw [physicalRecoveredPath_actual]
rfl
theorem physicalBranch_midpoint_raw {N K M G : ℕ} {L : Length}
(hL : L.val=(K:ℝ)) (hK : 1≤K) (h50 : 50<L.val)
(μ : Measure (BridgeEndpoints N L)) [SFinite μ]
(g : Fin G) (b : Fin N→Bool) (a : GridAllocation N K (G*M))
(m : ℕ) (r R C S s₀ t : ℝ) (u : Time) :
((physicalBranchLaw hL μ g b a m r R C S s₀ (2*t) u).compProd
(datumGroupReference hL g a m r R C S t u)).map
(fun z ↦ (z.1,physicalRecoveredMidpoint g a (z.1,projectGroup z.2)))=
(rawPhysicalBranch hL μ g b a m r R C S s₀ (2*t) u).map
(fun z ↦ (physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a),
labelBridgePath N L (2*t) z midpointTime)) := by
have hm := measurable_fst.prodMk (physicalRecoveredMidpoint_measurable (L:=L) g a)
have H := congrArg (fun ν ↦ ν.map (fun z ↦ (z.1,physicalRecoveredMidpoint g a z)))
(physicalBranch_reference_raw hL hK h50 μ g b a m r R C S s₀ t u)
rw [datumPhysicalReference,Measure.compProd_map projectGroup_measurable,
Measure.map_map hm (measurable_id.prodMap projectGroup_measurable)] at H
have hr : Measurable (fun z : BridgeEndpoints N L×(Fin N→Bridge.Noise) ↦
(physicalDatum g m r R C S s₀ u (labelBridgePath N L (2*t) z,a),
fun i : {i // i∈groupUnion g a} ↦ onePathAt i (labelBridgePath N L (2*t) z))) :=
((physicalDatum_measurable g m r R C S s₀ u).comp
((labelBridgePath_measurable N L (2*t)).prodMk measurable_const)).prodMk
(Measurable.of_eval fun i ↦ (onePathAt_continuous (i:Fin N)).measurable.comp
(labelBridgePath_measurable N L (2*t)))
rw [Measure.map_map hm hr] at H
simpa only [Function.comp_def,Prod.map,Function.id_def,physicalRecoveredMidpoint_actual] using H
end HardSphere.Groups
end
end
end OAI
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