| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.PhysicalVertex |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital |
|
|
| def datumCollisionBound (t r h R C : ℝ) : ℝ≥0∞ := |
| (ENNReal.ofReal (9/10:ℝ)*(ENNReal.ofReal (99/100:ℝ)*midpointLower t 13))⁻¹ * |
| (2*ENNReal.ofReal (hitConstant*(4*C*(2*t)*r+8*r*(1+R)^3/Real.sqrt (h/2)))) |
|
|
| theorem datum_reference_collision {N K M G m : ℕ} {L : Length} |
| (hL : L.val=(K:ℝ)) (h50 : 50 < L.val) (hm : 0 < m) |
| (g : Fin G) (t r R C S h : ℝ) (u : Time) |
| (ht : 0<t) (hr : 0<r) (hr1 : r≤1) (hR : 0≤R) (hC : 0≤C) (hh : 0<h) |
| (hu : (u:ℝ)=h/(2*t)) (hsmall : r*R≤1) |
| (hτr : (2*t)/(m:ℝ)≤r^2) (hmscale : (m:ℝ)*r^2≤2*(2*t)) |
| (hwindow : 2*((2*t)/(m:ℝ))≤h) (hlarge : 8*(2*t)≤(m:ℝ)*h) (hdrift : 26/(m:ℝ)≤r) |
| (d : PhysicalDatum N K M G L) (i j : Fin N) |
| (hi : datumAccessible hL g m t r R C S u i d) |
| (hy : ‖datumStart hL d i-anchorPoint (datumHome hL d i)‖≤3) |
| (hz : ‖datumEnd hL d i-anchorPoint (datumHome hL d i)‖≤3) |
| (hsep0 : r+2*S<torusDistance L (projectPoint L (datumStart hL d i)) (projectPoint L (datumStart hL d j))) |
| (hsep1 : r+2*S<torusDistance L (projectPoint L (datumEnd hL d i)) (projectPoint L (datumEnd hL d j))) |
| (f : Prod.mk d ⁻¹' datumPermitted hL g m r R C S u j) : |
| referenceOn t (datumStart hL d i) (datumEnd hL d i) |
| (Prod.mk d ⁻¹' datumPermitted hL g m r R C S u i) |
| {p | TrialCollision r (projectCurve L f.val) p.val}≤datumCollisionBound t r h R C := by |
| let A := Prod.mk d ⁻¹' datumTarget hL g m t r R C S u i 0 |
| let E := Prod.mk d ⁻¹' datumPermitted hL g m r R C S u i |
| have hA : MeasurableSet A := (datumTarget_measurable hL g m t r R C S u i 0).preimage measurable_prodMk_left |
| have hE : MeasurableSet E := (datumPermitted_measurable hL g m r R C S u i).preimage measurable_prodMk_left |
| have hvol : ENNReal.ofReal (99/100:ℝ)≤volume A := hi 0 |
| have hA0 : volume A≠0 := |
| ((by norm_num : (0:ℝ≥0∞)<ENNReal.ofReal (99/100:ℝ)).trans_le hvol).ne' |
| have hAf : volume A≠⊤ := ne_top_of_le_ne_top (by rw [unitTarget_volume]; exact ENNReal.one_ne_top) |
| (measure_mono (show A⊆unitTarget (anchorPoint (datumHome hL d i)+targetStep 0) from fun _ h ↦ h.1)) |
| have hmem : ∀ᵐU ∂targetUniform A,U∈A := Measure.ae_smul_measure (ae_restrict_mem hA) _ |
| have hs : ∀ᵐU ∂targetUniform A,ENNReal.ofReal (9/10:ℝ)≤ |
| trialKernel t (datumStart hL d i) (datumEnd hL d i) U E := hmem.mono (fun _ h ↦ h.2) |
| rw [referenceOn_collision t r _ _ hE] |
| have H := individual_collision_bound hm (datumStart hL d i) (datumEnd hL d i) |
| (anchorPoint (datumHome hL d i)) (projectCurve L f.val) u h50 ht hr hr1 hR hC hh hu |
| hy hz f.property.1 hsmall hτr hmscale hwindow hlarge hdrift |
| (by simpa only [projectCurve,ContinuousMap.coe_mk,f.property.2.2.1.1] using hsep0) |
| (by simpa only [projectCurve,ContinuousMap.coe_mk,f.property.2.2.1.2] using hsep1) |
| hA hA0 hAf (by norm_num : (0:ℝ)≤13) |
| (fun U hU ↦ midpoint_bound _ _ _ U (hy.trans (by norm_num)) (hz.trans (by norm_num)) |
| (targetCell_radius _ 0 U hU.1)) E hE (fun _ h ↦ ⟨h.1,h.2.1⟩) hs |
| refine H.trans ?_ |
| refine mul_le_mul' ?_ le_rfl |
| apply ENNReal.inv_le_inv.mpr |
| exact mul_le_mul' le_rfl (mul_le_mul' hvol le_rfl) |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
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|
